  This work defines two classes of processes, that we term {\it tempered
fractional multistable motion} and {\it tempered multifractional stable
motion}. They are extensions of fractional multistable motion and
multifractional stable motion, respectively, obtained by adding an exponential
tempering to the integrands. We investigate certain basic features of these
processes, including scaling property, tail probabilities, absolute moment,
sample path properties, pointwise H\"{o}lder exponent, H\"{o}lder continuity of
quasi norm, (strong) localisability and semi-long-range dependence structure.
These processes may provide useful models for data that exhibit both dependence
and varying local regularity/intensity of jumps.
