  We study how the "full hierarchy" of invariant manifolds for nonautonomous
integrodifference equations on the Banach spaces of $p$-integrable functions
behaves under spatial discretization of Galerkin type. These manifolds include
the classical stable, center-stable, center, center-unstable and unstable ones,
and can be represented as graphs of $C^m$-functions. For kernels with a
smoothing property, our main result establishes closeness of these graphs in
the $C^{m-1}$-topology under numerical discretizations preserving the
convergence order of the method. It is formulated in a quantitative fashion and
technically results from a general perturbation theorem on the existence of
invariant bundles (i.e.\ nonautonomous invariant manifolds).
