Diagonal platforms(stairs)

Questions
1

I am trying to implement digonal platform(stairs) for platformer.
But i have problem. When player move from platform to down stairs, he fall down(watch gif), but i need player to move down.
How can i do it? I think about raycast in the end of character, and on ray cast responce, make a decision is it normal fall down or i should down stairs

2

I think that raycasting is the way to go. Make sure that you have a solid way of casting the rays if there are tiny gaps in the physics.