  We compare two definitions of multistable L\'evy motions. Such processes are
extensions of classical L\'evy motion where the stability index is allowed to
vary in time. We show that the two multistable L\'evy motions have distinct
properties: in particular, one is a pure-jump Markov process, while the other
one satisfies neither of these properties. We prove that both are
semimartingales and provide semimartingale decompositions.
