That’s a fairly standard modern frontal takedown. You can actually see a pretty good example of how it works in Collateral.
Pushes are a very common technique, and variants appear in most martial arts. Some involve stepping on the foot, hooking behind the knee or ankle, or knocking a foot, or knee out of position. Then stepping through the space occupied by the opponent to send them sprawling.
I’m generalizing, but the takedown you’re describing is a simple and easy variant.
Once you put someone on the ground, the field is pretty open. Stomping on someone’s face isn’t subtle, but it is effective. Kicks to anything sensitive or important, like the genitals or head are another option. The specific takedown you’re talking about allows the attacker to simply keep walking after it’s completed, so while the victim is getting back on their feet, the attacker could slip into the crowd, or through a door.
For extra help:
Practical Unarmed Combatives: Low Line Destructives and the Secrets of the Silat Triangle This video by self-defense expert Michael Janich does a lot of explaining about how footwork affects pushing about six minutes in. This is important to understanding how to take someone down, the common assumption is that it’s strength. It’s not, it’s all about balance and unbalancing your opponent. There are many different ways to do it, but understanding the base concept will help you when figuring out how to write the scenario.