Complementarities are known to jeopardize stability in many-to-one matching markets. The present work explores a modeling approach tailored for large markets characterized by dynamic allocation with temporary partnerships and search frictions – the prominent example being labour markets – with the hope to obtain less stringent stability requirement. Weaker notions of stability of the process locally in time and asymptotically in market size are defined, according to which equilibrium may be obtained even when a stable equilibrium may not exist on the complete market. A general reduction method is proposed to study stability under general constraint graphs, in particular the category of graphs produced by a sequential search process. Application of this method to a simple search process yields an overall negative results : local stability on simple structures is sufficient, but symetrically local instability can spread to the whole graph. Asymptotic results may be recovered for large markets in which tree graphs are obtained ; it remains extremely challenging to ensure stability in general settings without any assumptions on the production function. Several qualitative properties of such models are explored, notably its (lack of) convergence to a static stable allocation when there exists one and optimality. In the particular case where the usual gross substitutes assumption is satisfied, the model should converge towards the static stable equilibria. This is, however, not the case in general even when the surplus function is submodular and there exists a stable equilibrium. Well known examples of instability are also revisited. Finally, what this approach can and cannot explain highlights several intuitions that might prove fruitful for further studies of stability issues with complementarities.