Let $Φ'$ denote the strong dual of a nuclear space $Φ$ and let $C_{\infty}(Φ')$ be the collection of all continuous mappings $x:[0,\infty) \rightarrow Φ'$ equipped with the topology of local uniform convergence. In this paper we prove sufficient conditions for tightness of probability measures on $C_{\infty}(Φ')$ and for weak convergence in $C_{\infty}(Φ')$ for a sequence of $Φ'$-valued processes. We illustrate our results with two applications. First, we show the central limit theorem for local martingales taking values in the dual of an ultrabornological nuclear space. Second, we prove sufficient conditions for the weak convergence in $C_{\infty}(Φ')$ for a sequence of solutions to stochastic partial differential equations driven by semimartingale noise.