A general class of finite variance critical $(\xi, \Phi,
k)$-superprocesses $X$ in a Luzin space $E$ with cadlag right Markov
motion process $\xi$, regular local branching mechanism and branching
functional $k$ of bounded characteristic are shown to continuously depend
on $(\Phi, k)$. As an application we show that the processes with a classical
branching functional $k(ds) = \varrho_s (\xi_s) ds [that is, a branching
functional $k$ generated by a classical branching rate $\varrho_s (y)] are
dense in the above class of $(\xi, \Phi, k)$-superprocesses $X$. Moreover,
we show that, if the phase space $E$ is a compact metric space and $\xi$
is a Feller process, then always a Hunt version of the $(\xi, \Phi,
k)$-superprocess $X$ exists. Moreover, under this assumption, we even get
continuity in $(\Phi, k)$ in terms of weak convergence of laws on Skorohod path
spaces.