On the other hand, housing them and/or giving them money would seem as a “reward”.

You’re framing the issue from a peculiar angle: Your question appears to be “how can we hurt refugees in the most humane way?”. What’s the end goal of that? Racial purity? A socially stagnant, shrinking society?

Imo the question should be: “how can we build a functioning society, especially knowing that people will continue to flee?”.

If you waste resources on hunting down and hurting the 95+% of people that don’t make any trouble, you’re almost by definition missing the people you do want to focus on, who will hurt society overall.