diff --git a/src/Bridges/bridge_optimizer.jl b/src/Bridges/bridge_optimizer.jl
index 7599be8b02..3949f148e4 100644
--- a/src/Bridges/bridge_optimizer.jl
+++ b/src/Bridges/bridge_optimizer.jl
@@ -320,6 +320,41 @@ function MOI.supports(
     return MOI.supports(b.model, attr)
 end
 
+function MOIU.pass_nonvariable_constraints(
+    dest::AbstractBridgeOptimizer,
+    src::MOI.ModelLike,
+    idxmap::MOIU.IndexMap,
+    constraint_types,
+    pass_cons;
+    filter_constraints::Union{Nothing,Function} = nothing,
+)
+    not_bridged_types = eltype(constraint_types)[]
+    bridged_types = eltype(constraint_types)[]
+    for (F, S) in constraint_types
+        if is_bridged(dest, F, S)
+            push!(bridged_types, (F, S))
+        else
+            push!(not_bridged_types, (F, S))
+        end
+    end
+    MOIU.pass_nonvariable_constraints(
+        dest.model,
+        src,
+        idxmap,
+        not_bridged_types,
+        pass_cons;
+        filter_constraints = filter_constraints,
+    )
+    return MOIU.pass_nonvariable_constraints_fallback(
+        dest,
+        src,
+        idxmap,
+        bridged_types,
+        pass_cons;
+        filter_constraints = filter_constraints,
+    )
+end
+
 function MOI.copy_to(mock::AbstractBridgeOptimizer, src::MOI.ModelLike; kws...)
     return MOIU.automatic_copy_to(mock, src; kws...)
 end
diff --git a/src/Test/modellike.jl b/src/Test/modellike.jl
index b1bc9e82d8..9d51eda101 100644
--- a/src/Test/modellike.jl
+++ b/src/Test/modellike.jl
@@ -783,7 +783,7 @@ function start_values_test(dest::MOI.ModelLike, src::MOI.ModelLike)
     end
 end
 
-function copytest(dest::MOI.ModelLike, src::MOI.ModelLike)
+function copytest(dest::MOI.ModelLike, src::MOI.ModelLike; copy_names = false)
     @test MOIU.supports_default_copy_to(src, true) #=copy_names=#
     MOI.empty!(src)
     MOI.empty!(dest)
@@ -850,13 +850,15 @@ function copytest(dest::MOI.ModelLike, src::MOI.ModelLike)
         MOI.Zeros,
     )
 
-    dict = MOI.copy_to(dest, src, copy_names = false)
+    dict = MOI.copy_to(dest, src, copy_names = copy_names)
 
-    @test !MOI.supports(dest, MOI.Name()) || MOI.get(dest, MOI.Name()) == ""
+    dest_name(src_name) = copy_names ? src_name : ""
+    @test !MOI.supports(dest, MOI.Name()) ||
+          MOI.get(dest, MOI.Name()) == dest_name("ModelName")
     @test MOI.get(dest, MOI.NumberOfVariables()) == 4
     if MOI.supports(dest, MOI.VariableName(), MOI.VariableIndex)
-        for vi in v
-            MOI.get(dest, MOI.VariableName(), dict[vi]) == ""
+        for i in eachindex(v)
+            MOI.get(dest, MOI.VariableName(), dict[v[i]]) == dest_name("var$i")
         end
     end
     @test MOI.get(
@@ -908,17 +910,17 @@ function copytest(dest::MOI.ModelLike, src::MOI.ModelLike)
     @test (MOI.VectorAffineFunction{Float64}, MOI.Zeros) in loc
 
     @test !MOI.supports(dest, MOI.ConstraintName(), typeof(csv)) ||
-          MOI.get(dest, MOI.ConstraintName(), dict[csv]) == ""
+          MOI.get(dest, MOI.ConstraintName(), dict[csv]) == dest_name("csv")
     @test MOI.get(dest, MOI.ConstraintFunction(), dict[csv]) ==
           MOI.SingleVariable(dict[w])
     @test MOI.get(dest, MOI.ConstraintSet(), dict[csv]) == MOI.EqualTo(2.0)
     @test !MOI.supports(dest, MOI.ConstraintName(), typeof(cvv)) ||
-          MOI.get(dest, MOI.ConstraintName(), dict[cvv]) == ""
+          MOI.get(dest, MOI.ConstraintName(), dict[cvv]) == dest_name("cvv")
     @test MOI.get(dest, MOI.ConstraintFunction(), dict[cvv]) ==
           MOI.VectorOfVariables(getindex.(Ref(dict), v))
     @test MOI.get(dest, MOI.ConstraintSet(), dict[cvv]) == MOI.Nonnegatives(3)
     @test !MOI.supports(dest, MOI.ConstraintName(), typeof(csa)) ||
-          MOI.get(dest, MOI.ConstraintName(), dict[csa]) == ""
+          MOI.get(dest, MOI.ConstraintName(), dict[csa]) == dest_name("csa")
     @test MOI.get(dest, MOI.ConstraintFunction(), dict[csa]) ≈
           MOI.ScalarAffineFunction(
         MOI.ScalarAffineTerm.([1.0, 3.0], [dict[v[3]], dict[v[1]]]),
@@ -926,7 +928,7 @@ function copytest(dest::MOI.ModelLike, src::MOI.ModelLike)
     )
     @test MOI.get(dest, MOI.ConstraintSet(), dict[csa]) == MOI.LessThan(2.0)
     @test !MOI.supports(dest, MOI.ConstraintName(), typeof(cva)) ||
-          MOI.get(dest, MOI.ConstraintName(), dict[cva]) == ""
+          MOI.get(dest, MOI.ConstraintName(), dict[cva]) == dest_name("cva")
     @test MOI.get(dest, MOI.ConstraintFunction(), dict[cva]) ≈
           MOI.VectorAffineFunction(
         MOI.VectorAffineTerm.(
diff --git a/src/Utilities/cachingoptimizer.jl b/src/Utilities/cachingoptimizer.jl
index 58c2b4b4a0..4c721ddebc 100644
--- a/src/Utilities/cachingoptimizer.jl
+++ b/src/Utilities/cachingoptimizer.jl
@@ -207,7 +207,8 @@ function _standardize(d::IndexMap)
 end
 
 function MOI.copy_to(m::CachingOptimizer, src::MOI.ModelLike; kws...)
-    return automatic_copy_to(m, src; kws...)
+    m.state == ATTACHED_OPTIMIZER && reset_optimizer(m)
+    return MOI.copy_to(m.model_cache, src; kws...)
 end
 function supports_default_copy_to(model::CachingOptimizer, copy_names::Bool)
     return supports_default_copy_to(model.model_cache, copy_names)
diff --git a/src/Utilities/copy.jl b/src/Utilities/copy.jl
index 8b73d4cd40..74995ac7af 100644
--- a/src/Utilities/copy.jl
+++ b/src/Utilities/copy.jl
@@ -422,6 +422,62 @@ function copy_constraints(
     end
 end
 
+function pass_nonvariable_constraints_fallback(
+    dest::MOI.ModelLike,
+    src::MOI.ModelLike,
+    idxmap::IndexMap,
+    constraint_types,
+    pass_cons = copy_constraints;
+    filter_constraints::Union{Nothing,Function} = nothing,
+)
+    for (F, S) in constraint_types
+        cis_src = MOI.get(src, MOI.ListOfConstraintIndices{F,S}())
+        if filter_constraints !== nothing
+            filter!(filter_constraints, cis_src)
+        end
+        # do the rest in `pass_cons` which is type stable
+        pass_cons(dest, src, idxmap, cis_src)
+    end
+end
+
+"""
+    pass_nonvariable_constraints(
+        dest::MOI.ModelLike,
+        src::MOI.ModelLike,
+        idxmap::IndexMap,
+        constraint_types,
+        pass_cons = copy_constraints;
+        filter_constraints::Union{Nothing,Function} = nothing,
+    )
+
+For all tuples `(F, S)` in `constraint_types`, copy all constraints of type
+`F`-in-`S` from `src` to `dest` mapping the variables indices with `idxmap`.
+If `filter_constraints` is not nothing, only indices `ci` such that
+`filter_constraints(ci)` is true are copied.
+
+The default implementation calls `pass_nonvariable_constraints_fallback` which
+copies the constraints with `pass_cons` and their attributes with `pass_attr`.
+A method can be implemented to use a specialized copy for a given type of
+`dest`.
+"""
+function pass_nonvariable_constraints(
+    dest::MOI.ModelLike,
+    src::MOI.ModelLike,
+    idxmap::IndexMap,
+    constraint_types,
+    pass_cons = copy_constraints;
+    filter_constraints::Union{Nothing,Function} = nothing,
+)
+    return pass_nonvariable_constraints_fallback(
+        dest,
+        src,
+        idxmap,
+        constraint_types,
+        pass_cons;
+        filter_constraints = filter_constraints,
+    )
+end
+
 function pass_constraints(
     dest::MOI.ModelLike,
     src::MOI.ModelLike,
@@ -473,13 +529,20 @@ function pass_constraints(
         (F, S) for (F, S) in MOI.get(src, MOI.ListOfConstraints()) if
         F != MOI.SingleVariable && F != MOI.VectorOfVariables
     ]
+    pass_nonvariable_constraints(
+        dest,
+        src,
+        idxmap,
+        nonvariable_constraint_types,
+        pass_cons;
+        filter_constraints = filter_constraints,
+    )
     for (F, S) in nonvariable_constraint_types
         cis_src = MOI.get(src, MOI.ListOfConstraintIndices{F,S}())
         if filter_constraints !== nothing
             filter!(filter_constraints, cis_src)
         end
         # do the rest in `pass_cons` which is type stable
-        pass_cons(dest, src, idxmap, cis_src)
         pass_attributes(dest, src, copy_names, idxmap, cis_src, pass_attr)
     end
 end
diff --git a/src/Utilities/model.jl b/src/Utilities/model.jl
index 8b8c6e859e..f9c7aaa0dc 100644
--- a/src/Utilities/model.jl
+++ b/src/Utilities/model.jl
@@ -752,6 +752,24 @@ function MOI.empty!(model::AbstractModel{T}) where {T}
     return
 end
 
+function pass_nonvariable_constraints(
+    dest::AbstractModel,
+    src::MOI.ModelLike,
+    idxmap::IndexMap,
+    constraint_types,
+    pass_cons;
+    filter_constraints::Union{Nothing,Function} = nothing,
+)
+    return pass_nonvariable_constraints(
+        dest.constraints,
+        src,
+        idxmap,
+        constraint_types,
+        pass_cons;
+        filter_constraints = filter_constraints,
+    )
+end
+
 function MOI.copy_to(dest::AbstractModel, src::MOI.ModelLike; kws...)
     return automatic_copy_to(dest, src; kws...)
 end
diff --git a/src/Utilities/universalfallback.jl b/src/Utilities/universalfallback.jl
index 892e6bfc17..e23bfcf611 100644
--- a/src/Utilities/universalfallback.jl
+++ b/src/Utilities/universalfallback.jl
@@ -86,6 +86,41 @@ function MOI.empty!(uf::UniversalFallback)
     return
 end
 
+function pass_nonvariable_constraints(
+    dest::UniversalFallback,
+    src::MOI.ModelLike,
+    idxmap::IndexMap,
+    constraint_types,
+    pass_cons;
+    filter_constraints::Union{Nothing,Function} = nothing,
+)
+    supported_types = eltype(constraint_types)[]
+    unsupported_types = eltype(constraint_types)[]
+    for (F, S) in constraint_types
+        if MOI.supports_constraint(dest.model, F, S)
+            push!(supported_types, (F, S))
+        else
+            push!(unsupported_types, (F, S))
+        end
+    end
+    pass_nonvariable_constraints(
+        dest.model,
+        src,
+        idxmap,
+        supported_types,
+        pass_cons;
+        filter_constraints = filter_constraints,
+    )
+    return pass_nonvariable_constraints_fallback(
+        dest,
+        src,
+        idxmap,
+        unsupported_types,
+        pass_cons;
+        filter_constraints = filter_constraints,
+    )
+end
+
 function MOI.copy_to(uf::UniversalFallback, src::MOI.ModelLike; kws...)
     return MOIU.automatic_copy_to(uf, src; kws...)
 end
diff --git a/test/Utilities/copy.jl b/test/Utilities/copy.jl
index 5ac74e9afb..4e45549330 100644
--- a/test/Utilities/copy.jl
+++ b/test/Utilities/copy.jl
@@ -45,7 +45,8 @@ end
     MOIT.failcopytestia(model)
     MOIT.failcopytestva(model)
     MOIT.failcopytestca(model)
-    MOIT.copytest(model, MOIU.Model{Float64}())
+    MOIT.copytest(model, MOIU.Model{Float64}(), copy_names = false)
+    MOIT.copytest(model, MOIU.Model{Float64}(), copy_names = true)
 end
 @testset "Allocate-Load" begin
     @test !MOIU.supports_allocate_load(DummyModel(), false)
@@ -55,7 +56,8 @@ end
     MOIT.failcopytestia(mock)
     MOIT.failcopytestva(mock)
     MOIT.failcopytestca(mock)
-    MOIT.copytest(mock, MOIU.Model{Float64}())
+    MOIT.copytest(mock, MOIU.Model{Float64}(), copy_names = false)
+    MOIT.copytest(mock, MOIU.Model{Float64}(), copy_names = true)
 end
 
 struct DummyEvaluator <: MOI.AbstractNLPEvaluator end
@@ -620,3 +622,73 @@ end
         MOI.NumberOfConstraints{MOI.SingleVariable,MOI.Integer}(),
     ) == 0
 end
+
+# We create a `OnlyCopyConstraints` that don't implement `add_constraint` but
+# implements `pass_nonvariable_constraints` to check that this is passed accross
+# all layers without falling back to `pass_nonvariable_constraints_fallback`
+# which calls `add_constraint`.
+
+struct OnlyCopyConstraints{F,S} <: MOI.ModelLike
+    constraints::MOIU.VectorOfConstraints{F,S}
+    function OnlyCopyConstraints{F,S}() where {F,S}
+        return new{F,S}(MOIU.VectorOfConstraints{F,S}())
+    end
+end
+MOI.empty!(model::OnlyCopyConstraints) = MOI.empty!(model.constraints)
+function MOI.supports_constraint(
+    model::OnlyCopyConstraints,
+    F::Type{<:MOI.AbstractFunction},
+    S::Type{<:MOI.AbstractSet},
+)
+    return MOI.supports_constraint(model.constraints, F, S)
+end
+function MOIU.pass_nonvariable_constraints(
+    dest::OnlyCopyConstraints,
+    src::MOI.ModelLike,
+    idxmap::MOIU.IndexMap,
+    constraint_types,
+    pass_cons;
+    filter_constraints::Union{Nothing,Function} = nothing,
+)
+    return MOIU.pass_nonvariable_constraints(
+        dest.constraints,
+        src,
+        idxmap,
+        constraint_types,
+        pass_cons;
+        filter_constraints = filter_constraints,
+    )
+end
+
+function test_pass_copy(::Type{T}) where {T}
+    F = MOI.ScalarAffineFunction{T}
+    S = MOI.EqualTo{T}
+    S2 = MOI.GreaterThan{T}
+    src = MOIU.Model{T}()
+    x = MOI.add_variable(src)
+    fx = MOI.SingleVariable(x)
+    MOI.add_constraint(src, T(1) * fx, MOI.EqualTo(T(1)))
+    MOI.add_constraint(src, T(2) * fx, MOI.EqualTo(T(2)))
+    MOI.add_constraint(src, T(3) * fx, MOI.GreaterThan(T(3)))
+    MOI.add_constraint(src, T(4) * fx, MOI.GreaterThan(T(4)))
+    dest = MOIU.CachingOptimizer(
+        MOI.Bridges.full_bridge_optimizer(
+            MOIU.UniversalFallback(
+                MOIU.GenericOptimizer{T,OnlyCopyConstraints{F,S}}(),
+            ),
+            T,
+        ),
+        MOIU.AUTOMATIC,
+    )
+    MOI.copy_to(dest, src)
+    voc = dest.model_cache.model.model.constraints.constraints
+    @test MOI.get(voc, MOI.NumberOfConstraints{F,S}()) == 2
+    @test !haskey(dest.model_cache.model.constraints, (F, S))
+    @test MOI.get(dest, MOI.NumberOfConstraints{F,S2}()) == 2
+    @test haskey(dest.model_cache.model.constraints, (F, S2))
+end
+
+@testset "copy of constraints passed as copy accross layers" begin
+    test_pass_copy(Int)
+    test_pass_copy(Float64)
+end