I don't buy it, sorry.

The poorest countries have the most generous welfare systems. Just laser-targeted to the elite. 

(Western countries also direct much direct and indirect welfare to the elite and their dependents.)

I propose that open borders are self-limiting in themselves, not via the proposed interaction with welfare. No one here agrees with my migration policies - except immigrants, who mostly don't want what they left behind to follow them. A critical mass of immigrants voting is the most certain and perhaps the only good way to end mass immigration, IMHO, YMMV