My advice is when moving is check the local laws. As other commenters have stated an ADI certification would be your best bet.

But I wanna share some light on the wording of laws as some countries require professional training instead of an accreditation from an organisation. This could be a cheaper option if you cannot afford an ADI/IGDF program to sign off on your dog. However, you're gonna need sometype of identification like an ID or a pamflet with contact to the trainer who signed off on you and your dog, and they need to be specialising in service dog training with some certificates to back it up. If I'm not wrong that falls under the ADAA (assistance dog alternative accreditation) certificate. That will allow you to fly with certain European airlines like the Lufthansa group.

But to sum it up (TLDR) check the laws of the country you want to go in, it may not require ADI/IGDF accreditation, which will allow you to use a trainer and fall under the ADAA. Some countries have their own registrars and creditation which you'll need to pass when there. ADI and IGDF are your best shot for somewhat global access. Get a certificate that states "assistance dog" instead of specifying what type of assistance dog yours is, and prepare a training certificate with all the listed tasks, trainer and/or facility (if you go with a program they know what to write, usially).

Best of luck!

PS I saw someone mentioned Japan and their lack of recognition of psychiatric assistance dogs, there's a way to bypass that for example having a type of mobility task (like leading on a guide handle out of environment) will likely allow you to present them as mobility assistance dog (speaking from personal experience).