  We analyze variants of the contact process that are built by modifying the
percolative structure given by the graphical construction and develop a robust
renormalization argument for proving extinction in such models. With this
method, we obtain results on the phase diagram of two models: the Contact
Process on Dynamic Edges introduced by Linker and Remenik and a generalization
of the Renewal Contact Process introduced by Fontes, Marchetti, Mountford and
Vares.
