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How do I move something that is not the default pawn so that it has the same ability to climb up?

So what I want is like when you have a Third Person project and the character steps up onto the stairs instead of just stopping there or something. I want to be able to do this with things derived from classes other than the default pawn.

The default character has a character movement component. The height of the steps the character
can travel is controlled by MaxStepHeight.

You can look at the source code where that is used and try to figure out how it works yourself.

However - not from the source itself - the regular way to do these things is to check the location you’re moving to before the move actually happens, and if the delta height is lower than max step height, you “lift” the character (or pawn) up the height of the floor it was about to move to.

You will need to create your own Movement and Input Component from scratch for this.
take a look at UCharacterMovement Component it’s a bit complicated due to networking client side prediction.

You can either go with full on Physics based controller, which uses forces to move or do it with a kinematic approach. other ways might be using Traces to see if something blocks the player than move only up to the hit location.

On Side note: take a look at unity forums and tutorials as well for making movement from scratch.