We consider the problem of allocating indivisible items to agents where both agents and items are partitioned into disjoint groups. Following previous works on public housing allocation, each item (or house) belongs to a block (or building) and each agent is assigned a type (e.g. ethnicity group). The allocation problem consists in assigning at most one item to each agent in a good way while respecting diversity constraints. Based on Schelling's seminal work, we introduce a generic individual utility function where the welfare of an agent not only relies on her preferences over the items but also takes into account the fraction of agents of her own type in her own block. In this context, we investigate the issue of stability, understood here as the absence of mutually improving swaps, and we define the cost of requiring it. Then we study the behaviour of two existing allocation mechanisms: an adaptation of the sequential mechanism used in Singapore and a distributed procedure based on mutually improving swaps of items. We first present the theoretical properties of these two allocation mechanisms and we then compare their performances in practice through an experimental study.