The controller tracks a floor face that it is currently 'on'. If no face is set, when the collision detection runs, if an object returns a separation vector with a Y value greater than a threshold, a ray is cast from the base of the capsule to see if it is directly underneath. If it is, the length of the ray from the capsule to the nearest face is stored, and the face is set to be the current floor face.
Each loop, if a face is set, a check is made to see if the distance from the capsule base to the face is greater than this distance. If so, the floor face is cleared and the object is considered to be in the air again.
Gravity is only applied each loop if there is no current floor face set.
The next part involved googling around for some maths to produce a quaternion representing the rotation from one vector to another. The movement vector is produced in response to velocity as if it is flat across a plain. But when a floor face is set, I compute a quaternion from (0, 1, 0) to the normal of the floor face, then extract a rotation matrix from this quaternion.
Then, when a floor face is current, the movement vector is just transformed by this matrix (the white cross in the image shows some test code to ensure this worked properly) and the capsule moves along the slope at the same speed it would have moved along a flat plain.
A small amount of tolerance when checking the distance is required to allow for smooth passing from one shape to another but generally this seems to work perfectly. Next up, I need to figure out how to have a maximum slope limit but we are certainly getting there.