Calling Emigrators: Transitioning to PE AND moving overseas - one at a time?

Hey monkeys. I've been on the buy side as an equity analyst for just over 3 years now, after changing careers (from astronomy, PhD under the belt). I'm based in South Africa and wishing to emigrate. I have a British passport, but trying to break into finance in another country still feels daunting - I barely know where to start! Further to that my long game is to transition to PE, so my questions are: should I try get a PE job in my home country first? With the idea that the skilling up will make getting a PE job overseas easier. Is it harder to get a PE job (in most Western countries) than in the listed space? Should I rather move first to the desired country in the same role I have now then try look for a PE job while I'm there?

Any help/guidance - including details of good European/US recruiters who recruit international - would be greatly appreciated!

Thanks