Let $\xi_{\varepsilon}(t)$ , $t\geq 0$ , be a continuous from the right stochastic process without discontinuities of the second kind and $\nu_{\varepsilon}$ , for each $\varepsilon\geq 0$ , be a non-negative random variable. In the paper, general sufficient conditions are studied for weak convergence of the distribution functions of the random variables$\xi_{\varepsilon}(\nu_{\varepsilon})$to the distribution function of$\varepsilon_0(\nu_0)$as$\varepsilon\to 0$.