How do you transfer stocks over to another platform e.g Saxo market to tiger brokers or moomoo? Can you even do that?

It is definitely possible to transfer your positions between brokers. However do take note of the costs involved. Both the outgoing and incoming brokerage will change you fees. If your position size is small, maybe it would be better for you to just sell of the stock and buy in using another broker.

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If you have a relatively large position size, or do not want to miss being "out of the market" due to settlement timings, maybe you can look into brokerage firms that provide a waiver of the transfer fees. Personally I have not done such transfer before, hence I cannot advise on the administrative process.

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My personal opinion is that there is no point in constantly chasing the brokerage with the lowest fees. There is a huge price war going on right now so you can expect multiple changes in fees from all the different competitors. As long as the fees your broker charge is affordable, and you trust and understand how the platform works, you should be fine.