  In this paper we prove that, under the assumption of quasi-transitivity, if a
branching random walk on ${{\mathbb{Z}}^d}$ survives locally (at arbitrarily
large times there are individuals alive at the origin), then so does the same
process when restricted to the infinite percolation cluster
${{\mathcal{C}}_{\infty}}$ of a supercritical Bernoulli percolation. When no
more than $k$ individuals per site are allowed, we obtain the $k$-type contact
process, which can be derived from the branching random walk by killing all
particles that are born at a site where already $k$ individuals are present. We
prove that local survival of the branching random walk on ${{\mathbb{Z}}^d}$
also implies that for $k$ sufficiently large the associated $k$-type contact
process survives on ${{\mathcal{C}}_{\infty}}$. This implies that the strong
critical parameters of the branching random walk on ${{\mathbb{Z}}^d}$ and on
${{\mathcal{C}}_{\infty}}$ coincide and that their common value is the limit of
the sequence of strong critical parameters of the associated $k$-type contact
processes. These results are extended to a family of restrained branching
random walks, that is, branching random walks where the success of the
reproduction trials decreases with the size of the population in the target
site.
