Having trouble trying to implement a mechanic in my 2D platformer game

Hello. For a couple of weeks, I've been trying to implement a mechanic in my side-scroller 2D Platformer game in which when you find an ally, it starts following behind you through the level. It may sound simple, but there are a lot of things that need to be taken into consideration like making sure that after jumping over a gap and staying next to the edge, the ally doesn't fall into the pit, the fact that it needs to be time-based, not frame-based and also some way of making the whole mechanic look smooth. Can you help me?