Isn’t an Open Immigration Policy consistent with Free Trade?

I was just reading an interesting post by Dr. Mark Thoma about illegal immigration and unemployment where he discusses Daniel Indviglio’s article. This right after reviewing some of Dr. Don Boudreaux’s comments on trade with China which lead me to wonder….

By closing our borders to open immigration we are in fact not allowing free trade. Is there a difference between a Chinese product (say a circuit board) used in the production of a good or service here (say a cash register) and an hour’s labor from a Mexican immigrant? They both seem to be factors of production.

One could argue that the illegal immigrant would take up jobs intended for Americans. The same can be said for the circuit board.

Politically I wonder if Republicans want to stifle free trade as much as they want to stifle immigration.

I am aware that immigration is a hot topic and I am firmly for open borders unlike many Americans. I believe, and someday hope to write a persuasive essay on the subject; that opening the borders would cause short-term stress but will establish in the long-term, peace and prosperity and equilibrium across borders for all.

In fact, I believe that the laws of supply and demand works for citizens as well as other goods. If we were to open the borders with Mexico we might have a huge influx of Mexicans seeking a better life. In the short run we would endure significant adjustments to the U.S. labor marketplace. However, in the long run Mexico will either endure a shortage of citizens (workers, entrepreneurs) or Mexican markets will find a way to entice them (or immigrants from another country) back to Mexico. Mexicans and others will move into an improved standard of living here and eventually back in Mexico as both country’s labor markets come around to an open border equilibrium.

My question is, are we being hypocritical to advocate free trade without free immigration?