This question is a follow-up from a previous question Self-deportation of American Citizens from US, and the purpose of my question is not to discuss the particulars of the Galicia event, but to ask the broader question about the actual process of transporting a person from the US to Mexico (or for that matter any other country).
It occurs to me that there are two major categories of deportation: (1) Where the CBP/ICE are in custody of a person and with force put that person on a plane,train, bus. etc that will only discharge that passenger in a targeted country and (2) where the person agrees to self-deport, in which case, they are not in custody and promise to arrange their own transportation out of the country (where they go and how they get there is their choice).
Focused on the first category, where CBP/ICE has custody of the person, does (for example Ice Airways) just fly into Mexico or Guatemala City and dump people off on the tarmac without notice? Or does the receiving country have to agree to accept 'the human cargo'? Can the receiving country refuse admission (particularly where the receiving country is unconvinced that the person being deported is actually a citizen of the receiving country)?
So the question I am asking is what is: What are the mechanics of deportation both in the US as well as in the "receiving" country?
(I am NOT asking about the process before a final deportation order is issued) or even person elects to "self-deport", rather I'm asking about what happens after)