If one is interested only in finding the minimal cost, then transfer-matrix method over the tropical semiring $R$ will do the job. That is, consider the graph weighted adjacency matrix over $R$, raise it to the power $l-1$, and "sum up" (in $R$) the entries corresponding to the pairs of indices from $S\times T$.

If one is interested only in finding the minimal cost, then transfer-matrix method over the tropical semiring $R$ will do the job. Namely, consider the line digraph $G:=L(D)$ and its weighted adjacency matrix over $R$, raise it to the power $l-2$, and "sum up" (in $R$) the entries corresponding to the pairs of possible start and end edges in $G$ (corresponding to pairs of vertices from $S\times T$ in $D$) multiplied by their costs.