Yes they are supposed to be under the direction of each states governor. The president can “federalize” them and then utilize them.

Another issue is the intent to use them in other states where those states governor’s haven’t given permission.

Basically the president can federalize a state’s national gaurd and direct them within that state. That’s legal, but is supposed to only be used during emergencies. Last time this was done was to prevent violence during the civil rights movement. However, moving them to another state without that state’s permission is likely illegal as is using the national gaurd as a long term police force

They are under the direction of each state’s governor, but could be mobilized into duty by the President under specific circumstances. In fact, many units were sent to Iraq and Afghanistan during those conflicts.

It’s unclear, though, whether the President has any ability to direct troops from one state to conduct activities in another. That may require the governor’s of both states to actually do the mobilization. However, there are a bunch of Republican states who would be happy exchanging troops to help arrest brown people.