From the point of view of stochastic analysis the Caputo and Riemann- Liouville derivatives of order α ε (0, 2) can be viewed as (regularized) generators of stable Lévy motions interrupted on crossing a boundary. This interpretation naturally suggests fully mixed, two-sided or even multidimensional generalizations of these derivatives, as well as a probabilistic approach to the analysis of the related equations. These extensions are introduced and some well-posedness results are obtained that generalize, simplify and unify lots of known facts. This probabilistic analysis leads one to study a class of Markov processes that can be constructed from any given Markov process in R^d by blocking (or interrupting) the jumps that attempt to cross certain closed set of 'check-points'.