We study the evolution of a  particle system whose genealogy is given by
a supercritical  continuous time Galton-Watson tree.  The particles move
independently according to  a Markov process and when  a branching event
occurs, the offspring locations depend on the position of the mother and
the  number of  offspring.  We prove  a  law of  large  numbers for  the
empirical  measure of  individuals alive  at time  t. This  relies  on a
probabilistic interpretation  of its intensity  by mean of  an auxiliary
process. This latter has the  same generator as the Markov process along
the branches plus additional  branching events, associated with jumps of
accelerated rate and biased distribution.  This comes from the fact that
choosing an  individual uniformly  at time t  favors lineages  with more
branching events and larger  offspring number. The central limit theorem
is  considered  on  a  special  case. Several  examples  are  developed,
including   applications  to   splitting  diffusions,   cellular  aging,
branching L´evy processes and ancestral lineages.