Can RUI grants be transferred to research universities? Suppose a NSF RUI (research at undergraduate institute) proposal is funded. Can the funding be transferred to a research university if the PI move to that university?
Sometimes the funding can be moved. Specifically, both university need to approve the transfer.  The official program details note:


  (i) Policy. When a PI/PD plans to leave an organization during the course of a grant, the organization has the prerogative to nominate a substitute PI/PD or request that the grant be terminated and closed out. In those cases where the PI/PD's original and new organizations agree, NSF will facilitate a transfer of the grant and the assignment of remaining unobligated funds to the PI/PD's new organization. This should normally be done with a tripartite agreement (involving NSF, the PI/PD’s original organization and new organization), or by a subaward arrangement between the PI/PD's original and new organizations, subject to NSF's consent. (See Chapter VII.B.3)


There are other details on the NSF webpage as well. I would encourage you to read that page if you have an RUI grant and this situation applies to you.