  In this work, convergence of evolving Finslerian metrics first in a general
flow next under Finslerian Ricci flow is studied. More intuitively it is proved
that a family of Finslerian metrics $g(t)$ which are solutions to the
Finslerian Ricci flow converge in $C^{\infty}$ to a smooth limit Finslerian
metric as $ t $ approaches the finite time $ T $. As a consequence of this
result one can show that in a compact Finsler manifold the curvature tensor
along Ricci flow blows up in short time.
