Pushing is Leshrac’s forte, so basically in the later stages of the game, whenever you have some kind of advantage over your opponents (for example, you have killed one or two of them), you should push. As Lesh does this very quickly, often you can snatch a tower before your opponents can set up a proper defense.
5-man Pushing
If you are pushing together with your team while your opponents are alive, you need to make the decision whether you want to stay close to the tower to deal damage with Edict. Bear in mind that in the mid and late game your carries likely have more than enough damage to bring a tower down, so your Edict might not be that important - you, being alive, on the other hand, might be very important for the upcoming fight.
It is a fact that the tower will fall faster with the help of your Edict. Despite that, if you are squishy, you will be a very good target to initiate on and to kill instantly. So, the main decision whether to stay on the front lines and deal damage to the tower or to stay behind your team more safely comes from two things.
Firstly, if you don’t expect any defense (most likely because you have a number advantage or you are trading), you should deal as much damage to the tower as possible (this means, as explained before, positioning yourself in such a way that the Edict deals damage only to the tower or simply killing all enemy units before that so that they can’t tank up Edict damage).
Secondly, if you expect that a fight might happen, you should consider how tanky you are, and what will happen if the enemy team initiates on you. As a support Lesh, it is better to stay behind your team. This way you will be able to help with a crucial Stun and the damage from your spells after the enemies initiate. If you are a farming Lesh, however, you can usually afford to stay at the front lines and deal damage to the tower. You should be tanky enough to survive the initiation and deal damage with your Pulse Nova in the middle of the fight. Bear in mind that if you manage to switch the Nova on before you get disabled, it will deal damage even while you are disabled. This might make initiating on you instead of your allies actually something positive.
Split-Pushing
As a farming Lesh, you could clear waves and push towers on your own incredibly fast. The problem is, however, that you have no escape mechanism, and if you get caught by a gank, you are likely to die. This is why split pushing with Lesh could be very rewarding but also very risky. Generally, you should do it only if you see the whole enemy team (or at least all heroes that can kill you) somewhere on the map (this way there will be no risk from getting ganked). This will give you an opportunity to split push and force the enemy team to react (by TPing to defend). If they don’t, you might get a free tower.
Anti-Pushing
Lightning Storm is one of the most potent spells for stalling pushes and defending towers without initiating. Provided you cast it on the enemy creep wave, you will damage both the creeps AND the enemy heroes that are nearby, and you will draw the creep aggro. The best part is that in 4 seconds you will be able to do this again.