That's until the authorities turn their eyes on Bitcoin. Border control could spot the most obvious methods of storing coins, like crypto apps on phone, hardware wallets, unencrypted flash drives, but another problem is what to do with bitcoin once you are in foreign country. Exchanging btc for fiat in a foreign country in a p2p way could be risky for a refugee, who in case of Ukraine are only women and children.
Phone apps? I've never seen them demand to unlock a phone for them. This is an invasion of privacy. Anyway, spotting a crypto wallet is close to impossible because you can have a thumb drive or a memory card with you and they'd have to spend hours searching through it all to find if you have a wallet installed or not. You could also have a simple text file with your private key with a changed extension, so that it looks like a system library or something.
The only way for people to start losing their stuff is if the border guards would start simply stealing all electronics: phones, memory cards, cameras, smart watches... Then there's always a way to smuggle files on a sim card, put it inside your shoe or something. In extreme situations you can always send a password protected file to someone you know in the country to which you're trying to flee, or put it in the cloud. If you know they're going to strip search you and take everything, cloud actually looks safer.
I've seen some of you talk about currency exchange. You won't even be able to exchange anything. I've heard that Ukrainians who moved to Poland have trouble exchanging their fiat because local exchanges don't want to accept it anymore in fear of the government collapse in Ukraine. They don't want to be stuck with extremely volatile UAH or RUB.
I thought that having cash is pretty safe, but it appears that you need to have "safe" cash like USD, GBP, CHF, or similar. If you hold on to your Lebanese or Sudanese pounds, you may end up holding a bunch of paper one day.