This paper establishes various variational properties of parametrized versions of two convexity-preserving constructs that were recently introduced in the literature: the proximal composition of a function and a linear operator, and the integral proximal mixture of arbitrary families of functions and linear operators. We study in particular convexity, Legendre conjugacy, differentiability, Moreau envelopes, coercivity, minimizers, recession functions, and perspective functions of these constructs, as well as their asymptotic behavior as the parameter varies. The special case of the proximal expectation of a family of functions is also discussed.