  In this paper we introduce non-decreasing jump processes with independent and
time non-homogeneous increments. Although they are not L\'evy processes, they
somehow generalize subordinators in the sense that their Laplace exponents are
possibly different Bern\v{s}tein functions for each time $t$. By means of these
processes, a generalization of subordinate semigroups in the sense of Bochner
is proposed. Because of time-inhomogeneity, two-parameter semigroups
(propagators) arise and we provide a Phillips formula which leads to time
dependent generators. The inverse processes are also investigated and the
corresponding governing equations obtained in the form of generalized variable
order fractional equations. An application to a generalized subordinate
Brownian motion is also examined.
