Do free traders neglect immigration?

At the end of a post about Social Security, Dean Baker implies free traders are lax in supporting freer immigration to the United States. At the risk of taking his closing quip too seriously, here’s why I am not fully persuaded.

Most economists who support freer trade also support freer immigration. Heck, Jagdish Bhagwati even has a version of his biography tailored to focus on immigration. And 500 economists signed a petition supporting more open immigration. So, although they could always do better, I think free traders are fulfilling their responsibility of advocating freer immigration. (One reason their contribution may be underappreciated is that economics is less central to the debate about immigration policy than trade policy.)

Then there are non-economist free traders. Here, Baker’s case holds up better. For example, the Heritage Foundation has used the argument that trade and migration are substitutes as a reason to support trade deals. On the other hand, Cato’s free traders are staunchly pro-immigration.

Two questions:

(1) Can anyone name an economist who is a notable free trader but opposes more liberal immigration policy?
(2) Are economists devoting too much of their policy influence to trade and not enough to migration?