Historic Home Help Needed

Many of my friends know I grew up in a historic home in downtown Stone Mountain. This house was made out of granite from the mountain and georgia red clay as the outside "finish". Unfortunately, being red clay it is eroding off of the house and it's really starting to need help. I have no idea how to fix it (or to stop it), so I thought I'd put this out and see if anyone else has ideas. What we have to keep in mind is that it can't be covered up (no stucco or paneling), so likely it has to be sealed. I thought maybe a polyurethane sealant might work, but I'm not positive.

This is where it's most obvious. The front door. The granite shouldn't be visible.

Gap between the front door and the wall to the left of the door.

Gap between the wall and the door to the right.

Suggestions? Preferably without breaking the bank?