Left to their devices, minions will eventually be able to push even to the base turrets. Which side accomplishes this, however, is random, and it's basically based on how attacks are being distributed.
Minion targetting works on the following priority scheme, brought to you by the League of Legends Wiki:
- An enemy champion who is attacking an allied champion.
- An enemy minion who is attacking an allied champion.
- An enemy minion who is attacking an allied minion.
- An enemy turret who is attacking an allied minion.
- An enemy champion who is attacking an allied minion. (Tangential, but this lower priority is why they don't pay attention to you when they're bashing your towers)
- The closest enemy minion.
- The closest enemy champion.
Now, when waves engage each other, there are a lot of minions attacking a lot of minions. So each minion picks some sort of target, but it is mostly random within that scope. So blue's caster minions could all be focusing down one purple melee minion, while purple's caster minions are split with two on one blue caster and one on a blue melee. The focus fire will reduce the enemy numbers, in turn resulting in one side having an advantage. But because how the focus fire is attributed is random (both mechanically and because the actual positioning of the minions can differ greatly as a result), the side that gets an advantage is completely random.
This compounds due to two major factors as the game progresses.
- Once siege minions enter the fray, they both deal more damage and take substantially more damage. They can prolong battles long enough for secondary and tertiary minion waves to support them, which can greatly turn the tide of battle even more depending again on how focus fire is attributed.
- If a small minion wave pushes far enough, then when the next minion wave comes in the enemy will have a shorter travel time before its support shows up. With the aforementioned siege minion stalling, the end result is that one side builds up a massive force that just has to deal with the opposing side one wave at a time.
As such, eventually, they amass enough force to push down full towers, much harder than the defense can manage. Even when the defense can stall, the sheer numbers built up from the initial push can have a devastating advantage... at least until their focusing messes up and the force builds up in the opposite side.