Cut off the incentives and they will self-deport. Start with strict border enforcement. Then, speed up the process by adding incentives such as $1000 payments to be deposited in a country of their destination and a free plane ride home for the first million or so to apply for these free plane rides and waive their rights. Gradually reduce the payment until it is zero, but they get a free bus ride home. Then, finally, crank it up to full prosecution, hard jail time, finger printing, disqualification from future entry and asset seizures if they insist on dragging it out. The carrot and the stick. Take your pick.
As is turning off the spigot of refugee resettlement and chain migration of relatives of those here.