T = 5;
dim = 3;

dKmat = Array[dKm, {T, T}];
M1mat = Array[M1m, {dim, dim}];
Kron3 = KroneckerProduct[M1mat, dKmat];

ArrayReshape[Kron3, {Times @@ Dimensions[Kron3], 1}] ===
 KroneckerProduct[
   Flatten[Table[
     KroneckerProduct[IdentityMatrix[T, SparseArray],
      Transpose[{M1mat[[;; , i]]}]], {i, dim}], 1],
   IdentityMatrix[T, SparseArray]].ArrayReshape[
   dKmat, {Times @@ Dimensions[dKmat], 1}]