Is it common for a lab to pay for international moving expenses?
I assume you didn’t just walk in to the office one day and got told “pack your things! We’re moving to Austria!” Instead, my guess is you were asked if you were agreeable to the idea of moving, told under what terms, and agreed (well, perhaps with the alternative being that you would become unemployed, but still).
If my guess is correct, the time when you were asked would have been the best time to ask the question about relocation expenses. If you have already agreed to the move under terms that do not include covering your relocation expenses, then sure, you could bring it up now, but it’s possible that that ship has sailed; if the PI or receiving institution are stretched for cash, they have little actual incentive to be receptive to your wishes.
Bottom line: there is no “should” or “shouldn’t”. In the workplace, you commonly get what you have leverage to ask and can get people to agree to give you. It sounds like you don’t have that kind of leverage in the current situation. Perhaps your PI will be understanding and agree to your request, which is reasonable after all. But there is no rule or ethical principle that says what should happen.