  We consider a strong Markov process with killing and prove an approximation
method for the distribution of the process conditioned not to be killed when it
is observed. The method is based on a Fleming-Viot type particle system with
rebirths, whose particles evolve as independent copies of the original strong
Markov process and jump onto each others instead of being killed. Our only
assumption is that the number of rebirths of the Fleming-Viot type system
doesn't explode in finite time almost surely and that the survival probability
of the original process remains positive in finite time. The approximation
method generalizes previous results and comes with a speed of convergence. A
criterion for the non-explosion of the number of rebirths is also provided for
general systems of time and environment dependent diffusion particles. This
includes, but is not limited to, the case of the Fleming-Viot type system of
the approximation method. The proof of the non-explosion criterion uses an
original non-attainability of $(0,0)$ result for pair of non-negative
semi-martingales with positive jumps.
