Concrete – filling in gaps in concrete patio

set your door with available shimming it in place then use a 4000 psi rated NON SHRINK grout this will harden and will not allow the door to move .be sure its non shrink use a small trowel to push to grout under the jam don't try and get away with only using your hands the trial will help you make sure you have no air bubble so there will be no room for water to fill small voids and freeze causing separation good luck

From the two pictures, it looks like the porch is two parts - the foundation slab, and the top surface. Is the slab moving, or just the top surface?

My suggestion would be to apply low-expanding spray foam (they make a grey-colored version so it doesn't look ridiculous, as well as one with pest deterring chemicals) to those cracks. If it expands too far out of the top-surface crack, trim it after it's try, then apply caulk to the face of it to waterproof it.

Continue to monitor it for movement going forward. It's important to determine how and why you're getting movement.