We analyze the problem of matching asylum seekers to member states, incorporating wait times, preferences of asylum seekers, and the priorities, capacities, and burden-sharing commitments of member states. We identify a unique choice rule that addresses feasibility while balancing priorities and capacities. We examine the effects of both homogeneous and heterogeneous burdensizes among asylum seekers on the matching process. Our main result shows that when all asylum seekers are treated as having identical burden-sizes, the asylum-seeker-proposing cumulative offer mechanism guarantees both stability and strategy-proofness. In contrast, when burden-sizes vary, there are scenarios where achieving stability or strategy-proofness is no longer possible.