TTL (time to live) is how long a DNS answer stays cached. That is why record changes are not instant: usually minutes to a day, and up to 24–72 hours for a name server change.
Before moving a website, lower the TTL in advance so the switch-over is faster, then restore the usual value afterwards.
If some visitors see the old website and others the new one after a change, that is normal caching and will resolve itself.
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