  We give necessary and sufficient criteria for the Feller-Dynkin property of
solutions to martingale problems in terms of Lyapunov functions. Moreover, we
derive a Khasminskii-type integral test for the Feller-Dynkin property of
multidimensional diffusions with random switching. For one dimensional
switching diffusions with state-independent switching, we provide an
integral-test for the Feller-Dynkin property.
