You are correct of course that extra people coming into a profession could cause stagnation of wages, but it is usually far more complicated than that. In the UK, despite all of this immigration we have rather low rates of unemployment, marginally lower than before the 2004 expansion of the EU. This seems counterintuitive if wage growth is an over-supply problem. It is also not true to say that an industry will just keep paying more and more to attract workers from other sectors. This soon hits the point in the modern globalised world that the industry just becomes uncompetetive and closes down, outsources, invests in labour reducing machinery, or moves. Wage stagnation is very real in Britain however, and for pointers to the cause you might want to look at why non wage inflation has been so low for so long, putting aside the last 18 months of FX driven inflation caused by comparatively weak sterling post Brexit vote (This adds around 15% to most imported goods).
[0]:https://en.wikipedia.org/wiki/Freedom_of_movement_for_worker...