  We study the evolution of states of an infinite system of point particles
dwelling in a locally compact Polish space $X$. Each particle produces at
random a finite `cloud' of offsprings distributed over $X$ according to some
law, and disappears afterwards. The system's states are probability measures on
an appropriate space of locally finite counting measures on $X$. Their
evolution is obtained by solving the corresponding Fokker-Planck equation. We
prove that this equation has a unique solution and discuss some of its
properties. Our pivotal idea of dealing with infinite systems consists in
passing to tempered counting measures by imposing appropriate restrictions on
the branching. In this approach, we first solve a nonlinear evolution equation
in the space of bounded continuous functions on $X$ -- so called log-Laplace
equation. Next we solve the Kolmogorov equation which is then used to solve the
Fokker-Planck equation and thus describe the evolution in question.
