The Markov renewal equation \[ Z_i (x) = z_i(x) + \sum_{j \in \mathbb{E}} \int_0^x Z_j(x-y) F_{ij} (\mathrm{d} y), \qquad i \in \mathcal{E}, \] is considered in the subcritical case where the matrix of total masses of the \(F_{ij}\) has spectral radius strictly less than one, and the asymptotics of the \(Z_i(x)\) is found in the heavy-tailed case involving a local subexponential assumption on the \(F_{ij}\). Three cases occur according to the balance between the \(z_i(x)\) and the tails of the \(F_{ij}\), A crucial step in the analysis is obtaining multivariate and local versions of a lemma due to Kesten on domination of subexponential tails. These also lead to various results on tail asymptotics of sums of a random number of heavy-tailed random variables in models which are more general than in the literature.