Let \(\Sigma\), T, and \(\Theta\) be convergences on \(2^ X\) (the space of subsets of X), \(2^ Y\), and \(2^{X\times Y}\) (the space of multifunctions from X to Y), respectively. The authors are interested only in those convergences \(\Theta\) with the following property: if a filtered family \({\mathcal A}\) of sets converges to A in \((2^ X,\Sigma)\) and a filtered family \({\mathcal M}\) of multifunctions converges to M in \((2^{X\times Y},\Theta)\), then the family \({\mathcal M}{\mathcal A}\) converges to MA in \((2^ Y,T)\). This is the same as requiring the continuity of the evaluation map (A,M)\(\mapsto \cup_{x\in A}Mx\) from \((2^ X\times 2^{X\times Y},\Sigma \times \Theta)\) to \((2^ Y,T)\). In fact, each convergence \(\Theta\) studied in this paper is defined to be or is proved to be the coarsest such convergence for some particular \(\Sigma\) and T arising (in various ways) from the topologies on X and Y, respectively. Applications in Hadamard differentiation and in optimization theory are given.