  We study the $n$-component priority asymmetric simple exclusion process
($n$-ASEP) with reflecting boundaries. We obtain all invariant measures in
explicit form and prove reversibility. Using the symmetry of the generator of
the process under the quantum algebra $U_q[\mathfrak{gl}(n+1)]$ we construct
duality functions with respect to which the $n$-ASEP is self-dual, both for the
finite and the infinite integer lattice. For the $n$-ASEP on the infinite
lattice we use self-duality to derive in explicit form the time evolution of a
family of measures with $K$ shocks in terms of the transition probability of
$K$ coloured particles in a shock exclusion process with particle-dependent
hopping rates and nearest-neighbour colour exchange. This process is a gas of
particles that forms a bound state, corresponding to shock coalescence on
macroscopic scale.
