diff --git a/src/Utilities/copy.jl b/src/Utilities/copy.jl
index 76a34dc75f..8beb2cfada 100644
--- a/src/Utilities/copy.jl
+++ b/src/Utilities/copy.jl
@@ -383,6 +383,21 @@ function default_copy_to(dest::MOI.ModelLike, src::MOI.ModelLike)
         error("Model $(typeof(dest)) does not support copy_to.")
     end
     MOI.empty!(dest)
+    for (F, S) in MOI.get(src, MOI.ListOfConstraintTypesPresent())
+        if F == MOI.VariableIndex
+            if !MOI.supports_add_constrained_variable(dest, S)
+                throw(MOI.AddConstraintNotAllowed{F,S}())
+            end
+        elseif F == MOI.VectorOfVariables
+            if !MOI.supports_add_constrained_variables(dest, S)
+                throw(MOI.AddConstraintNotAllowed{F,S}())
+            end
+        else
+            if !MOI.supports_add_constrainet(dest, F, S)
+                throw(MOI.AddConstraintNotAllowed{F,S}())
+            end
+        end
+    end
     index_map, vis_src, constraints_not_added =
         _copy_variables_with_set(dest, src)
     # Copy variable attributes
@@ -498,8 +513,8 @@ function _add_variable_with_domain(
     src,
     index_map,
     f,
-    ci::MOI.ConstraintIndex{MOI.VariableIndex,<:MOI.AbstractScalarSet},
-)
+    ci::MOI.ConstraintIndex{MOI.VariableIndex,S},
+) where {S<:MOI.AbstractScalarSet}
     set = MOI.get(src, MOI.ConstraintSet(), ci)
     dest_x, dest_ci = MOI.add_constrained_variable(dest, set)
     index_map[only(f)] = dest_x
@@ -512,8 +527,8 @@ function _add_variable_with_domain(
     src,
     index_map,
     f,
-    ci::MOI.ConstraintIndex{MOI.VectorOfVariables,<:MOI.AbstractVectorSet},
-)
+    ci::MOI.ConstraintIndex{MOI.VectorOfVariables,S},
+) where {S<:MOI.AbstractVectorSet}
     set = MOI.get(src, MOI.ConstraintSet(), ci)
     dest_x, dest_ci = MOI.add_constrained_variables(dest, set)
     for (fi, xi) in zip(f, dest_x)
diff --git a/test/Utilities/copy.jl b/test/Utilities/copy.jl
index ce39a7e83d..b51624c51f 100644
--- a/test/Utilities/copy.jl
+++ b/test/Utilities/copy.jl
@@ -1003,6 +1003,35 @@ function test_copy_to_sorted_sets()
     return
 end
 
+function test_add_constraint_not_allowed_scalar_variable()
+    F, S = MOI.VariableIndex, MOI.GreaterThan{Float64}
+    model = MOI.Utilities.Model{Float64}()
+    x, _ = MOI.add_constrained_variable(model, MOI.GreaterThan(1.0))
+    dest = MOI.FileFormats.CBF.Model()
+    @test_throws(MOI.AddConstraintNotAllowed{F,S}, MOI.copy_to(dest, model))
+    return
+end
+
+function test_add_constraint_not_allowed_vector_variables()
+    F, S = MOI.VectorOfVariables, MOI.AllDifferent
+    model = MOI.Utilities.Model{Float64}()
+    x, _ = MOI.add_constrained_variables(model, MOI.AllDifferent(3))
+    dest = MOI.FileFormats.CBF.Model()
+    @test_throws(MOI.AddConstraintNotAllowed{F,S}, MOI.copy_to(dest, model))
+    return
+end
+
+function test_add_constraint_not_allowed_scalar_function()
+    F, S = MOI.ScalarNonlinearFunction, MOI.EqualTo{Float64}
+    model = MOI.Utilities.Model{Float64}()
+    x = MOI.add_variable(model)
+    f = MOI.ScalarNonlinearFunction(:log, Any[x])
+    MOI.add_constraint(model, f, MOI.EqualTo(0.0))
+    dest = MOI.FileFormats.CBF.Model()
+    @test_throws(MOI.AddConstraintNotAllowed{F,S}, MOI.copy_to(dest, model))
+    return
+end
+
 end  # module
 
 TestCopy.runtests()