How do I get collisions between Tilemap Collider 2d and a Kinematic Rigidbody 2d?

So, basically I'm making a top-down view, 2d game.
This means I don't want gravity, so I've put the rigidbody2d of my player character to be Kinematic. I move it based on mouseclicks, using RigidBody2D.MovePosition. Now, I have a tilemap, that consists of walls that you should not be able to move through. I have a Tilemap collider 2d on the tiles, and in scene view, I can see a nice green collider grid BUT my character can move through them no matter what I do.

So, what am I doing wrong here? I'm not sure I really understand how these colliders should work. If I set the rigidbody to dynamic, it works fine with the colliders, but then he drops, due to gravity (which I don't want).