We prove the existence of solutions for an evolution quasi-variational
inequality with a first order quasilinear operator and a variable convex set,
which is characterized by a constraint on the absolute value of the gradient
that depends on the solution itself. The only required assumption on the
nonlinearity of this constraint is its continuity and positivity. The method
relies on an appropriate parabolic regularization and suitable {\em a priori}
estimates. We obtain also the existence of stationary solutions, by studying
the asymptotic behaviour in time. In the variational case, corresponding to a
constraint independent of the solution, we also give uniqueness results
