We consider a sequence of Hilbert spaces and convex, lower semicontinuous functionals defined on them. Any such functional generates a gradient flow on the underlying space. In the case of a fixed space, it is known that convergence of the sequence of gradient flows is guaranteed by convergence of the functionals in the sense of Mosco. However, in many interesting cases, such as discrete-to-continuum limits, thin domains, or boundary layer problems, the underlying space changes along the sequence. We introduce a generalization of Mosco-convergence to such setting, based on a notion of "connecting operators". We prove a corresponding general weak convergence result for the flows, without assuming any coercivity of the sequence of functionals, and discuss some applications. Our main tool is the notion of "variational solutions" in the sense of Bögelein et al, which allows performing a "Gamma-convergence"-type argument in the evolutionary setting. This is joint work with Y. Giga and P. Rybka.