No one could really say what the absolute position of a union would be in that question. Unions are membership organizations that work in the interest of current, and to some extent future, members. The rest is politics.

A union would have incentive for foreigners to e.g. come to the US, so the US get a larger slice of the market. On the other hand they don't have an interest in those workers e.g. having lesser rights or lower compensation. Because then foreigner workers become more attractive than their existing members, thereby preventing those members from having long term success. So it depends on how you view e.g. H1B.

It really isn't that complicated if you think about it.

Do you have an example of such pro immigration labor union?

You are unlikely to find a western labor union that is specifically pro immigration these days. Since labor has been in decline in the west for the last few decades. Before that I am sure you could find those who were pro labor immigration. The Nordic countries have for example had relatively large labor immigration under social democratic governments (which are usually associated with the labor unions).

Also it depends what you mean by pro immigration. No union is going to be pro liberal immigration rules, since they don't want to undermine their own members. For it make sense for a union it has to be employees who are lacking, either in numbers or skill, in the country and can therefor expand the country's share of the market.

The IWW.

Why? Would the union only consist of American workers?

Because that's what unions do. They protect current members at the expense of everyone else, including unionized colleagues.