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 burden-sizes 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.