Flushing fixes your device, not the whole internet
Your local flush only clears the copy stored on your own computer or phone. Every DNS resolver between you and the destination -- your internet provider's resolver, and countless caching servers worldwide -- keeps its own copy until that entry's time-to-live (TTL) expires on its own schedule, which is why a change can look "fixed" on one device and "still broken" on another for a while.
Mobile devices don't have a dedicated flush button
Most phones do not expose a menu specifically for clearing the DNS cache the way a desktop command line does. Turning airplane mode off and on, or simply restarting the device, forces it to re-establish its network connection from scratch, which has much the same practical effect.
Frequently Asked Questions
Does flushing DNS slow down my internet afterward?
No noticeable slowdown. Only the very next visit to each site takes a few extra tens or hundreds of milliseconds while it looks up the address again -- after that it is cached and just as fast as before.
Can I flush the DNS cache on my phone?
There usually is not a dedicated menu option for it on smartphones. Toggling airplane mode off and on, or restarting the device, re-establishes the network connection and produces essentially the same result.