Hard to say. Kinematics aren't used for their collision responsiveness. Often that's why they're preferred, so they will ignore certain impacts. IE - Moving platforms.
If you're writing your own physics code, you have to figure out what all these situations are, study the outputs, and fine tune solutions to your design. There are all kinds of techniques, such as casting rays, using special collision shapes, etc. There is a lot of study, and various pros and cons, so I'm afraid there is no easy answer that will be the magic bullet there.
I'm fairly sure the standard thing is to cast a ray, and detect if it's a wall collision. Then ignore whatever movement is being sent, and maybe even perform a vector slide along the collision normal. Such as this: http://docs.godotengine.org/en/stable/tutorials/2d/kinematic_character_2d.html#problem
I could also suggest some kinematic character tutorial videos @ https://www.youtube.com/playlist?list=PLFt_AvWsXl0f0hqURlhyIoAabKPgRsqjz
It's for Unity, but it may give you some ideas about the techniques and considerations you need to tackle when writing kinematic characters.
I hope that helps.