The grants given out by the United States government can be funded from either state treasuries or the federal revenues, depending on the grant program as well as where the application was sent.
Grants issued for major programs such as reconstruction of cites after natural disasters are nearly always funded out of the federal revenues as the federal government takes responsibility for the welfare of the large number of people affected.
However, the grants that state governments issue to small companies or individuals are funded out of the state treasury.
This means that if you apply to the state government for a grant and the proposal is accepted, the money that you receive will be given to you from the state treasury.
However if, for some reason, you apply to the federal government for assistance, the money you will receive will most probably come from the federal revenues.
However it is also possible that the US federal government may transfer your grant into the hands of your state government, but this occurrence is very rare and it only takes place if you apply for changes to the original proposal.
In this scenario the grant is initially funded from the federal revenues, but after the transfer to the state government, the source of the funding will divert to state treasuries.
If you apply for a state grant and move to another state while the application is in process, your application will most probably be rejected, unless the project you are looking to start will still provide benefits to the previous state.
For example, if you are starting a waste recycling company; and after moving to another state you are still willing to provide services in the state where you applied for the grant, there is a better chance of the proposal getting accepted.