Guardsmen swear an oath joining the US Army when they become Guardsmen.Not unless on title 32 activation. Other than that, under state control. That is why it takes the Governor to activate for use in riots inside the state, or in times of other state emergency.
The possibility of federal service overseas is another reason why the National Guard are not the militia. The Constitution limits the federal role of the militia to enforcing the law, suppressing insurrection, and repelling invasion. None of those involve service outside US borders.
Also, militiamen have the right to take their weapons home with them. Guardsmen leave their weapons locked up on base when they go home.