I don't know if it's as simple as just rich countries not getting vaccines to poorer countries.
I have spoken to several people with families in poorer countries who say that while the vaccine is available there and free, people have to pay to be given the injection. Clearly it isn't OK for HCPs from richer countries to fly in and run vaccination clinics unasked, being seen to take jobs from the locals. But it does mean that where people can't pay, or would rather not, vaccines are going out of date rather than getting into arms.
I'm involved with a charity in another country. We have been trying to get our 'clients' vaccinated for months, but we just can't get it arranged.