We study how small perturbations of matrix polynomials may change their elementary divisors and minimal indices by constructing the closure hierarchy graphs (stratications) of orbits and bundles of matrix polynomial Fiedler linearizations. We show that the stratication graphs do not depend on the choice of Fiedler linearization which means that all the spaces of the matrix polynomial Fiedler linearizations have the same geometry (topology). The results are illustrated by examples using the software tool StratiGraph.