For sealing the gap, look into self-leveling caulks. See if they fit your needs.

I use Vulkem 45, a self-leveling polyurethane caulk. They also have a Vulkem 45SSL, which is a semi-self leveling. It's more viscous than the easier flowing self-leveling.

What you'll typically do is fill the gap with sand to about 1/4" below the surface, that way you're not wasting caulk by pouring too deep. If you had wider gaps, you could fill the gap with foam backer rod.

I pour the sand into the gap using a cup.

Have all your prep work done. This caulk is "self-leveling", so it will indeed flow. It's easier to caulk all in one shot versus starting and stopping.

If you get any overflow, just let the "puddle" of caulk cure overnight. Come back the next day with a utility knife and cut the overflow puddle from the fill line, then you can peel the overflow up off of the stone.