  This paper is devoted to the study of a fractional version of non-linear
$\mathpzc{M}^\nu(t)$, $t>0$, linear $M^\nu (t)$, $t>0$ and sublinear
$\mathfrak{M}^\nu (t)$, $t>0$ death processes. Fractionality is introduced by
replacing the usual integer-order derivative in the difference-differential
equations governing the state probabilities, with the fractional derivative
understood in the sense of Dzhrbashyan--Caputo. We derive explicitly the state
probabilities of the three death processes and examine the related probability
generating functions and mean values. A useful subordination relation is also
proved, allowing us to express the death processes as compositions of their
classical counterparts with the random time process $T_{2 \nu} (t)$, $t>0$.
This random time has one-dimensional distribution which is the folded solution
to a Cauchy problem of the fractional diffusion equation.
