  We study a system of branching Brownian motions on $\mathbb R$ with
annihilation: at each branching time a new particle is created and the leftmost
one is deleted. In [7] it has been studied the case of strictly local creations
(the new particle is put exactly at the same position of the branching
particle), in [10] instead the position $y$ of the new particle has a
distribution $p(x,y)dy$, $x$ the position of the branching particle, however
particles in between branching times do not move. In this paper we consider
Brownian motions as in [7] and non local branching as in [10] and prove
convergence in the continuum limit (when the number $N$ of particles diverges)
to a limit density which satisfies a free boundary problem when this has
classical solutions, local in time existence of classical solution has been
proved recently in [13]. We use in the convergence a stronger topology than in
[7] and [10] and have explicit bounds on the rate of convergence.
