Unvetted animal shelter volunteers should not be a problem for the animal rescue non-profits. The criteria for being a volunteer is assessed and the volunteer needs to do practical work and training too. This is before being allowed to take on any real responsibility as they should be accompanied by a trained volunteer, or staff member who can show them the ropes. Worldwide, the volunteer industry is unregulated with no global regulations that can be applied to volunteers, as each industry must have their regulations, ethics and standards that should apply to volunteers, as much as to full time staff.
If a volunteer applies to an animal shelter, and they are only in the country temporarily, this needs to be addressed, as how committed are they? Also, applications for volunteering should ask the hard questions. Including what the volunteer would do in hypothetical situations, have they ever been involved in a crime, have a record, or anything against their name that would stop them from being a good volunteer? The only way to not have problems with volunteers is to give them a trial period first, and then take on full time if proven suitable as an animal rescue volunteer.