Maybe it is just posturing and lies, but the conservative viewpoint has been to have a merit-based system for permitting immigrants into the country, as opposed to immigration based on familial relations. Every country gets to decide what kind of people that they want coming into their country and I think its perfectly fine for conservatives to feel that way (it is a different matter that I don't agree with that idea). My perception (again, which might be wrong) is that that is essentially what Republicans have been getting at for the past couple of decades (perhaps more).
Now, that being said: these are the same folks who were about Family Values, Fiscal Purity etc. So, like I said before, I don't think the arguments are being made in good faith. But the argument does have merit.
Go look up the stuff that Orin Hatch and Rubio sponsored with regards to H1B visas. The stuff that they sponsored would double the number of H1Bs and was majority sponsored by republicans.
further restrict currently-legal immigration
This has nothing to do with debates over executive authority to choose how to prioritize enforcement and to make transparent commitments regarding that (which the executive clearly has in some cases, outside the obvious pardon power, since literally every grant of immunity from prosecution, or plea bargain to reduced charges, is an enforceable commitment by the executive not to enforce some law in some well-defined circumstance based on policy priorities.)
Perhaps, but conservatives have frequently put forward bills (or, when in the executive, instituted executive policy without Congressional action) that further restrict currently-legal immigration and non-immigrant entry and privileges, which is more significant sign of opposition to legal immigration than merely voting against expansions to legal immigration, whether pure or not.