If you're thinking of women/couples having a lot of kids to collect those kids' welfare, it depends on how the welfare for those kids compares to the cost of raising the kid. Also many people would be morally against bringing a kid into a life of potential suffering just for their own monetary gain. But for someone with no dependents, moving to a new country poses no risks to anyone except themselves.
In that sense, the risk to oneself is so great that only a privileged few in developing countries who wanted to relocate would actually have the resources to overcome them. So for many people, relocating might legally be a choice, but it would be practically impossible.
Realistically, migration is only an option for you if you (1) have a job offer from a company based in the place you intend to migrate, (2) have sufficient funds + skills to cover yourself until you can find a job, (3) have family who can support you and lend you their networks until you can find employment, (4) have sufficient funds to start your own business. Incidentally, those criteria also form categories of US visas (H1-B/TN-1/E-3/L-1, O-1, Family Reunion, EB-5), though, they impose more stringent requirements than economics would.
The amount of a welfare state you can sustain is certainly dependent on your tax revenue, and in turn on the quality of your economy, on political willingness to set high enough taxes, etc. But I don't think there's a point at which a welfare state is flatly impossible.
And there are certainly lots of unforeseen possibilities. Perhaps an economy where employers can hire anyone without bureaucracy will grow companies that generate enough revenue, even without lots of human employees on payroll, to sustain the welfare state.