I am but an egg when it comes to minion wave control. These are some my observations thus far.

Under normal circumstances when your minion wave collides with the enemy minion wave this is the formation that results.

M=Melee
C=Caster
N=Cannon

CCC
MMM
---
MMM
CCC

or

CCC
N
MMM
---
MMM
N
CCC

There will be a gap between where the melee minions line up to fight each other and the ranged minions line up, unless the minions are reacting to your presence as they will preferentially target champions if they are in range.

So, one sure fire method to get your lane to push toward enemy territory is to kill the enemy ranged creeps in the next two minions waves. Doing this will cause your minion waves to stack (puts more than one minion wave together) because your waves will stop moving when they encounter the line of enemy melee minions that you left behind, which allows time for oncoming waves to catch up.*, **, ***, ****

So, if you have the time and want to make life easy on yourself, follow your minion wave so you don't disrupt the battle lines that form. Once the melee minions are fighting each other you can walk around them and kill off the caster minions.

The cannon minions do screw this up a bit, so to avoid damaging the melee minions, just roughly stand on the middle caster minion while dishing out damage to the ranged contingent of the minion wave.

*There are formulas out there for how many minions you need in your wave to expect them to push X distance into enemy territory, but I don't have them memorized and standing towers complicate matters.

**The point of creating a giant wave is to force the enemy team to deal with it our risk losing a tower or perhaps even an inhibitor, so as say Top Laner, it is good practice to setup your lane to push while you roam or Teleport in to help you team with an objective.

***If a giant wave from your side makes it to the enemies' outer base turret and doesn't kill the turret the enemy team will have a giant minion wave that counter pushes because their minions will not have far to travel from their Nexus to the outer base turret, which causes their waves to stack. So, be prepared to deal with that and if you have someone who is underfarmed or only a minion wave or two in gold from completing an major item, send them over to deal with giant wave. (if they are squishy or the enemy team has good catch potential just make sure to ward for them so they can farm the wave safely).

****Yes you really do need to NOT kill the melee minions if you want to create a giant minion wave. I know it is hard to leave that tasty tasty farm behind, but do it if you want this to work.