  The study of distributed order calculus usually concerns about fractional
derivatives of the form $\int_0^1 \partial^\alpha u \, m(d\alpha)$ for some
measure $m$, eventually a probability measure. In this paper an approach based
on L\'evy mixing is proposed. Non-decreasing L\'evy processes associated to
L\'evy triplets of the form $\l a(y), b(y), \nu(ds, y) \r$ are considered and
the parameter $y$ is randomized by means of a probability measure. The related
subordinators are studied from different point of views. Some distributional
properties are obtained and the interplay with inverse local times of Markov
processes is explored. Distributed order integro-differential operators are
introduced and adopted in order to write explicitly the governing equations of
such processes. An application to slow diffusions is discussed.
