You've reached late game in League of Legends, and are full build. Do you sell boots for more damage or sustain, or keep them for survivability? Well, you are not the first person to ask yourself this question.
The real answer is not always. Movement speed is an important stat and is the biggest necessity for kiting. If you’re playing someone with large amounts of base move speed, like a Viktor with phase rush and cosmic drive, I would sell my boots for a final full item.
If you’re playing someone that doesn’t come with lots of movement speed, say, Mao’kai, I wouldn’t sell my boots. You’ll never touch the enemy.
The other biggest factor is how much kiting you’ll have to do. If you need to kite out a bruiser, you should keep your boots. It’s no fun trying to run away from a Camille without movement speed.
For example, in a recent Tobias Fate stream, he bought 6 items as Gangplank and had an 80% crit chance, and got unlucky because the one barrel that would have won the game, didn't crit. Due to that, he ended up selling boots and picking up something else but then he died because he had no movement speed, and couldn't chase to finish people off.
As with all League of Legends guides, it really matters whom you are playing, and who you are playing against. There is a reason boots are a staple item in League of Legends, the same way flash is a staple summoner.