Let \(\alpha\), \(k\in L^1_{\text{loc}}([0,\infty))\) be two positive functions and let \(A\) be a densely defined closed linear operator in a Banach space \(Y\). The authors consider an \((\alpha,k)\)-regularized resolvent family \(\{R(t): t\geq 0\}\) on \(Y\) associated with a Volterra equation of convolution type involving the operator \(A\) (here, \(R(t)\) is a strongly continuous operator function on \(Y\) satisfying certain properties). Let \(X\) be another Banach space continuously embedded in \(Y\) and invariant for \(R(t)\). It is shown that the restriction of \(R(t)\) on \(X\) is strongly continuous with respect to the norm on \(X\) if and only if all its partial orbits are relatively weakly compact in \(X\). The authors then apply this result to integrated solution families, integrated semigroups and integrated cosine functions.