A generalization of the classical three-sided assignment market is considered, where value is generated by pairs or triplets of agents belonging to different sectors, as well as by individuals. For these markets we represent the situation that arises when some agents leave the market with some payoff by means of a generalization of Owen (Ann Econ Stat 25-26:71-79, 1992) derived market. Consistency with respect to the derived market, together with singleness best and individual anti-monotonicity, axiomatically characterize the core for these generalized three-sided assignment markets. When one sector is formed by buyers and the other by two different type of sellers, we show that the core coincides with the set of competitive equilibrium payoff vectors.