In this paper, we discuss a two-stage flow shop scheduling problem with batch processing machines. The jobs belong to different incompatible job families. Only jobs of the same job family can be batched together. The performance measure is the total weighted tardiness of the jobs. A decomposition heuristic is proposed that is based on the idea to iteratively determine due dates for the jobs in the first stage and earliest start dates of the jobs in the second stage. The two resulting subproblems are solved using a time window decomposition (TWD) heuristic and a variable neighborhood search (VNS) scheme. Results of computational experiments based on randomly generated problem instances are presented. We show that the VNS-based scheme outperforms the TWD heuristic. In addition, we show that the decomposition scheme can be parallelized in a very natural way. As a result, the amount of computing time is modest, even for the computational expensive VNS scheme.