"To process an adoption" means to go through the necessary steps and procedures to legally and officially transfer custody of a child or a pet from one set of parents or owners to another. Full definition
We make every effort to process adoption applications on the same day we received them (in person) based on a first come, best match applicant basis. (faithfulfriends.us)
As our daughter gets older, she is processing her adoption on a different level and feeling the loss of not being raised by her birth parents at times. (creatingafamily.org)
Usually, shelters are not as strict as rescues in processing adoptions but there can always be exceptions. (homeoanimal.com)