Yes, when it comes to making an embryo transfer, it does not matter if the embryos come from your own or donated eggs. As long as the recipient woman has regular menstrual and ovulatory cycles, it is possible to make an embryo transfer in a natural cycle.
However, it is not convenient to make a natural cycle if the transfer is going to be fresh, since it is very complicated to synchronize the donor with the recipient and, therefore, the probability of success would be lower.
As for the treatment of embryo adoption, this can also be done in a natural cycle and, furthermore, these embryos will always be cryopreserved.
Suggested for you: Embryo Transfer in Natural Cycles: Advantages & Disadvantages.