Free markets internationally absolutely include the free movement of labor, as well as the free movement of capital, goods and services, etc. And many services include labor.
Like most who only partially believe in free markets, you flake out on the most politically volatile element of free markets: the free movement of labor.
Prices cannot be set by supply and demand if the supply of labor is not as free to move as the other factors of production, and the finished products and services.
Nonsense. You can have free international trade (no tarfiffs) and still control which people (labor) you allow into your country. The two are orthogonal. On products you can also control which ones you allow into the country and which ones you don’t. Of course some flaky libertarians want zero controls on borders but they are idiots.