Yes, you definitely want to get the Kenya exit stamp in Nairobi. It's a simple visit to the Immigration office in downtown Nairobi.
I did the east side of Lake Turkana into Ethiopia in 2016, through Sibiloi Park, on motorbike with a buddy. Heading north, Loyangalani is the last sizable settlement you'll see for some time. Our route was through some rugged territory... you encounter a fair bit of deep sand, rock and dry bed river crossings... unless it's been raining, they they can be real river crossings.
Remoteness defines that route. There were a handful of Kenya park officers in Sibiloi Park, but at the southern end at Koobi Fora, a very basic campsite on the lake with rooms and maybe fuel and
if you're lucky. Preparation with ample fuel, water and food and fuel is important, and good GPS. If something went wrong inside the park, it could be many days before someone happened upon you, so prepare accordingly.
Once exiting Sibiloi at the north, you arrive at the village of Ileret. You can stop at the police station and "register"... but note, it's not a passport stamp, so be sure to get that in Nairobi. We camped at the police station, had a lot of fun.
From Ileret you're off to Ethiopia. In places it's just bush track, but GPS will get you north to the paved east-west road. Take that west to Omorate and check into the Ethiopia Immigration office with passport.
Overall, it's an amazing and unforgettable route, and you encounter many tribal peoples (though not inside Sibiloi itself). Challenging and rewarding in equal measure. I can't really speak to other routes, only the one I did.