diff --git a/exir/passes/constant_prop_pass.py b/exir/passes/constant_prop_pass.py
index 14ff651c936..764efffa18f 100644
--- a/exir/passes/constant_prop_pass.py
+++ b/exir/passes/constant_prop_pass.py
@@ -4,6 +4,8 @@
 # This source code is licensed under the BSD-style license found in the
 # LICENSE file in the root directory of this source tree.
 
+from typing import Callable, List, Optional
+
 import torch
 from torch._export.utils import get_buffer, get_param, is_buffer, is_param
 from torch._guards import detect_fake_mode
@@ -11,11 +13,27 @@
 from torch.export.exported_program import InputKind, InputSpec, TensorArgument
 
 
-def is_const(arg, exported_program, const_data_list) -> bool:
+_PRIMITIVE_TYPES = (
+    float,
+    int,
+    bool,
+    str,
+    torch.Tensor,
+    torch.device,
+    torch.dtype,
+    torch.layout,
+)
+
+
+def is_const(
+    arg: object, exported_program: ExportedProgram, const_data_list: List[str]
+) -> bool:
     if isinstance(arg, (tuple, list)):
         return all(is_const(x, exported_program, const_data_list) for x in arg)
     elif isinstance(arg, dict):
         return all(is_const(x, exported_program, const_data_list) for x in arg.values())
+    elif isinstance(arg, _PRIMITIVE_TYPES):
+        return True
     elif not isinstance(arg, torch.fx.Node) or arg.op != "placeholder":
         return False
     elif (
@@ -27,9 +45,11 @@ def is_const(arg, exported_program, const_data_list) -> bool:
     return False
 
 
-def get_data(exported_program, arg):
+def get_data(exported_program: ExportedProgram, arg):
     if isinstance(arg, (tuple, list)):
         return [get_data(exported_program, x) for x in arg]
+    elif isinstance(arg, _PRIMITIVE_TYPES):
+        return arg
     elif is_param(exported_program, arg):
         return get_param(exported_program, arg)
     elif is_buffer(exported_program, arg):
@@ -37,7 +57,10 @@ def get_data(exported_program, arg):
     return None
 
 
-def constant_prop_pass(exported_program: ExportedProgram) -> ExportedProgram:
+def constant_prop_pass(
+    exported_program: ExportedProgram,
+    skip_folding_node_fn: Optional[Callable[[torch.fx.Node], bool]] = None,
+) -> ExportedProgram:
     """
     This pass is for constant propagation for Exported Program with lifted parameters,
     as the parameters will not be shown up as `get_attr` but as `placeholder` to the graph.
@@ -56,12 +79,14 @@ def constant_prop_pass(exported_program: ExportedProgram) -> ExportedProgram:
     if len(has_cond) > 0:
         raise RuntimeError("constant_prop_pass for control flow is not supported yet.")
 
+    first_user_input_idx = -1
     first_user_input = None
-    for node in exported_program.graph.nodes:
+    for i, node in enumerate(exported_program.graph.nodes):
         if (
             node.op == "placeholder"
             and node.name in exported_program.graph_signature.user_inputs
         ):
+            first_user_input_idx = i
             first_user_input = node
             break
 
@@ -79,6 +104,9 @@ def constant_prop_pass(exported_program: ExportedProgram) -> ExportedProgram:
     assert fake_mode is not None
 
     for node in exported_program.graph.nodes:
+        if skip_folding_node_fn is not None and skip_folding_node_fn(node):
+            # Do not process this node if we were told to skip it.
+            continue
         if node.op == "call_function":
             constant_data_name_list = [
                 input_spec.target for input_spec in prop_constant_data
@@ -115,9 +143,11 @@ def constant_prop_pass(exported_program: ExportedProgram) -> ExportedProgram:
                     exported_program.state_dict[prop_constant_tensor_fqn] = (
                         prop_constant_tensor
                     )
-                    exported_program.graph_signature.input_specs.append(
-                        prop_constant_node_input_spec
+                    # Insert new buffers before the first user input.
+                    exported_program.graph_signature.input_specs.insert(
+                        first_user_input_idx, prop_constant_node_input_spec
                     )
+                    first_user_input_idx += 1
 
     # Remove the propogated buffer from the state dict
     for node in exported_program.graph.nodes:
@@ -128,6 +158,16 @@ def constant_prop_pass(exported_program: ExportedProgram) -> ExportedProgram:
         ):
             exported_program.state_dict.pop(node.name, None)
             exported_program.graph.erase_node(node)
+            # Delete the input spec for this deleted buffer.
+            to_erase_idx = []
+            for i, spec in enumerate(exported_program.graph_signature.input_specs):
+                if spec.arg.name == node.name:
+                    to_erase_idx.append(i)
+            assert (
+                len(to_erase_idx) == 1
+            ), f"Should only delete one spec per node, but deleting multiple: {to_erase_idx} {exported_program.graph_signature.input_specs}"
+            for i in reversed(to_erase_idx):
+                exported_program.graph_signature.input_specs.pop(i)
 
     exported_program.graph_module.recompile()
     return exported_program