  For a connected pasting scheme $\mathcal G$, under reasonable assumptions on
the underlying category, the category of $\mathfrak C$-colored $\mathcal
G$-props admits a cofibrantly generated model category structure. In this
paper, we show that, if $\mathcal G$ is closed under shrinking internal edges,
then this model structure on $\mathcal G$-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 $\mathcal G$ 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 $\mathcal G$-props. The final section gives illuminating examples
that justify the conditions on base model categories.
