  In this paper, we establish some functional central limit theorems for a
large class of general supercritical superprocesses with spatially dependent
branching mechanisms satisfying a second moment condition. In the particular
case when the state $E$ is a finite set and the underline motion is an
irreducible Markov chain on $E$, our results are superprocess analogs of the
functional central limit theorems of \cite{Janson} for supercritical multitype
branching processes. The results of this paper are refinements of the central
limit theorems in \cite{RSZ3}.
