  We find the number of compositions over finite abelian groups under two types
of restrictions: (i) each part belongs to a given subset and (ii) small runs of
consecutive parts must have given properties. Waring's problem over finite
fields can be converted to type~(i) compositions, whereas Carlitz and locally
Mullen compositions can be formulated as type~(ii) compositions. We use the
multisection formula to translate the problem from integers to group elements,
the transfer matrix method to do exact counting, and finally the
Perron-Frobenius theorem to derive asymptotics. We also exhibit bijections
involving certain restricted classes of compositions.
