Shrinkability, relative left properness, and derived base change

For a connected pasting scheme G, under reasonable assumptions on the underlying category, the category of C-colored G-props admits a cofibrantly generated model category structure. In this paper, we show that, if G is closed under shrinking internal edges, then this model structure on G-props satisfies a (weaker version) of left properness. Connected pasting schemes satisfying this property include those for all connected wheeled graphs (for wheeled properads), wheeled trees (for wheeled operads), simply connected graphs (for dioperads), unital trees (for symmetric operads), and unitial linear graphs (for small categories). The pasting scheme for connected wheel-free graphs (for properads) does not satisfy this condition. We furthermore prove, assuming G is shrinkable and our base categories are nice enough, that a weak symmetric monoidal Quillen equivalence between two base categories induces a Quillen equivalence between their categories of G-props. The final section gives illuminating examples that justify the conditions on base model categories.