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

Quasivariational solutions for first order quasilinear equations with gradient constraint