So-called "slight" decreases to wages add up to quite a lot of money over whole careers. I don't see any reason to deliberately institute a policy that helps advantage capital over labor, when it is already so very advantaged.
That would actually put labor on the same footing with capital, since capital can already move freely across borders, while labor cannot.
Hmmm I can invest my capital in 30 countries easily with a carefully selected index fund. If I could move across borders freely, it would be hard (if not impossible) to do the same thing. So I'd say it isn't the same footing that applies.
Because overall immigration does way more good than harm to a country, and that drawback effect on low wages can be countered by policy measures.