Investing newbie: Buy shares through one exchange and transfer to another?

Hi, I am a total investing newbie. I'm re-reading Invest Like a Pro for the 2nd time, the first time was in 2019... and we all know what 2020 was like.

I'm not based in the US, so my access to Index Funds and ETFs is limited. However, I see that I may be able to trade a little of the crypto I have for shares in ETFs on eToro. If I do that, can I then transfer these shares to something more mainstream like ETrade? Or are the investment purchases I make through one broker/exchange fixed to that broker/exchange?

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