  In this paper we study a flow by minkowskian curvature where we have a
different Minkowski plane at each time. We derive some evolution formulas,
present sufficient hypotesis for the short time existence and convexity of
solutions and study the motion considering a particular type of families of
Minkowski norms. Also, as a corollary we establish a result about a certain
family of nonlinear parabolic PDE's.
