Two libertarian arguments for having unrestricted immigration, here and here.
The first, at least, assumes no welfare benefits would be available to migrants.
Both worth reading in full, both persuasive, both uncomfortable to me. I am, though, persuaded by the argument (under the first link) that asking the state to act as a private individual might in limiting migration is a bad precedent:
First, to ask government to mimic the outcomes of a pure private property rights system is to come dangerously close to asking government to treat the entire country as if that country is the private property of the state. What an irony!
Anyone who advocates such a policy overlooks the single most important reason for strictly limiting government's power: Unlike true owners of private property, government can resort to force to increase the size of its property holdings and the value of its portfolio. Government is not an owner of private property. Restrictions on government discretion are appropriate precisely because government possesses a legitimized monopoly on coercion.
Consider, for example, the right of free speech. Would it be sensible to argue that, because each private-property owner has the right to regulate what is said on his property, government in our less-than-libertarian world should have the power to regulate speech uttered in public places or over public airwaves?