It really depends on the assignment clause in the lease and the terms of the assignment between the lessee and third party, but typically these assignments will be structured such that the original lessee is relieved of all obligations under the lease including the payment of royalties. In exchange for the assignment, the original lessee will often reserve a small overriding royalty which gets paid by the third party just like the MO's royalty.
If the original lessee participates as a working interest owner you mean? That would also depend on the duties it maintains under its assignment, but you can also add in your lease that to the extent asssigned, the original lessee will remain on the hook for any obligations, or that if asssigned, any assignee will assume all obligations under the lease.