What you say seems obvious is actually often untrue. More immigration /= lower wages all round. Not in the way most western democracies' economies are structured, anyway. So there is little to square.
I'll note that some answers here are referring to the American context, and accusing left leaning politicians of being less than genuine, that the immigrants are to work for substandard wages. In some countries, eg Australia, they crack down on anyone doing this, and it's not nearly as common as it might be in the states. The issue is worker protections and enforcement, more than it is increased immigration, if that is what you're seeing.