The phrase "surrender process" refers to the act of giving up or letting go of something, typically as a result of losing or accepting defeat. It means to willingly or reluctantly give in and stop fighting or resisting. Full definition
There is also a custodial surrender process for people who are taking care of other people's pets. (reference.com)
The pet surrender process starts by filling out the online appointment application. (waysidewaifs.org)
It's been a process of opening up, drawing upon the trust and surrender process of giving birth, and allowing that to carry over into our sexual relationship. (diahpodcast.com)