Yes. Governors can send troops to other states and they can recall them. This is very common. We helped in New Orleans, for example, in the recovery from the hurricane. And often this ends up being federally funded (especially when it is a natural disaster) as the federal government can send funds to states (not directly paying the troops, they are still title 32).
And the Defense Department with NGB could request that the Governors send troops to DC for security (which they did). The Governors do not have to send them, and if they do they can recall them. Since this is through NGB the state will be funded for the deployment, however the troops will be Title 32 under the authority of the Governor of their states.
BUT if the President calls up the National Guard (not requests, but federalizes the troops) then the Governor has no say so. He can neither deny the act or recall the troops because the National Guard belongs to the State and the Federal Government (the President is the Commander in Chief of the National Guard, as is the Governor).
The State Guard does not operate outside of the individual state. They are often called up by the Governor on a volunteer basis (they are not paid until they are actually working for the state) to man units when the National Guard is called up for federal service (for military deployments).
Governors can't send troops to other states unless the Governor of other states requests them. And that can be denied by the Governor.
The Defense Department, or the President can request. But the Governors can deny.
If the President 'federalizes' the Guard, then the Governor does have a say still. He, the President, can't federalilze and use the Guard against another state or for use in another state. Posse Comitatus Act. Even then, Governors still have a say. In 1980 Reagan sought to send the National Guard into Central America. Several Democratic Governors refused.
In 2020 several Democratic Governors refuse Trumps call to send troops, National Guard, to the states where rioting was taking place. And the Governor of Illinois warned Trump that no troops should be sent to his state.
In 2018, Governors of 8 states would refuse or recall Guard troops sent to protect the Border Wall with Mexico.
In other words, I am proud of Governor Abbott of Texas in recalling the Guard. In Hind sight, he never should have sent the Guard. Hopefully, these are lessons learned.
Quantrill