Fix Autopathing
Disclaimer: I don't know if autopathing is accountable for all the clunkiness in movement I'm seeing or if it's something else. However, my ping doesn't spike and I notice fewer problems if I hold right click in the directions of where I want to go. It could also be the game's weird collision detection system. I'm going ahead assuming the problem is League's autopathing system.
Autopathing keeps second guessing itself and making dumb corrections that waste time. If a unit changes course because of changing the destination or because of an obstacle, the champion will pause and then make a slight turn before proceeding. The game doesn't have turn speed so there's no reason for this happening.
I'd have clips of this if my computer were better but it isn't; there's no point showing 240p examples. I have observations and anecdotes instead.
- When an enemy is hiding on the other side of a tower and the enemy tries to maintain that distance, autopathing will make the player change directions and go around the long side of the tower's perimeter.
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When you're in the open space of your base and change your destination, your character makes two turns. The first turn is between where you were heading and where you want to go. Then there's a fidgeting moment where you don't move. Then your character heads towards your new destination. This is more noticeable with floating champions such as Thresh and Syndra.
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Minion block: In this case, it's the minions' pathing that's broken. The game decides that the minions around you should focus on attacking you instead of moving away so that they stop clipping into your character model. Your champion is confined in a space smaller than its collision radius and spazzes around trying to move. Ultimately, you're stuck unless you have a way to move through units.

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Monsters and straggler minions also get stuck randomly. Scuttle dashes into a wall and stops moving. A single minion will randomly start spinning around and not move. Rift Herald can do the same thing.
Whatever the problem actually is, make the game less clunky.