Not sure about the 360, but I moved from Europe to the US with a PS4 last year. Only bothered getting a new power cord and so far it hasn't blown up. As far as games go, yes they are region free, but DLC isn't (it's locked to the specific region store your PSN account is with). You wouldn't be able to, for example, buy a game locally in Mexico and then download DLC for it from the EU PlayStation Store - you'd need to open a local (MX? US?) PSN account.
You might as well take them, unless space is an issue and your girlfriend is arguing to pack Grandma's collection of bone china butter dishes or something. If they do end up breaking in Mexico City after some time, then you can buy new ones.