The biggest mistakes I have seen are:
1. Not investing enough to begin with. In many of our home countries, we pay high taxes, and in return for that get some kind of pension. Overseas, people typically need to invest more just to be where they would have been back at home.
2. Linked to the first point, there is a keeping up with the Jones' mentality in some of the larger expat cities like Dubai, Singapore, Shanghai etc. This is best avoided.
3. Only sending money back to home countries. This can be tax-inefficient . Most providers back home don't even accept expats, and lying about your residency is never a good idea. Take the UK as an example. If you live in the UK, you can invest in a tax-efficient investment ISA. You can't if you live overseas. If you try to use a bank statement as proof of address, and invest in an ISA, it is tax fraud. Usually more tax-efficient investments are available for expats.
4. Not having portable investments. Few investment platforms offer true portability if you move from country to country. This is essential for expats to avoid hassles and unexpected capital gains taxes.
5. Only investing in illiquid investments like property and businesses. Few people in the US, UK or Canada are interested in investing in Cambodian or Vietnamese real estate back home. They come to Asia, and suddenly "familiarity bias" means that they are more likely to do it, because "everybody else seems to be". I am not saying you should never buy property, but the risks involved in emerging market property is huge.
6. Giving money to a local wife or husband to buy land because the law makes it easier. This one is self-explanatory.
7. Neglecting wealth protection like health insurance. In our home countries, it isn't a must. In many overseas countries it is.
8. Lastly, making the same mistakes people tend to make back at home, like market timing, speculating etc.