We investigate existence and uniqueness of p-means and the median of a probability measure on a Finsler manifold, in relation with the convexity of the support of the measure. We prove that the p-mean is the limit point of a continuous time gradient flow. Under some additional condition which is always satisfied for larger than or equal to 2, a discretization of this path converges to the p-mean. This provides an algorithm for determining those Finsler center points.