From KAZ to Mongolia you must enter Russia. Normal Russian Visa rules apply.
In Russia you can enter a Border Zone on Federal roads (main roads) without further papers, but only if en-route to the border.
What we did.
This time last year, we joined the M52 half way between Gorno-Altaisk and Mongolia. We rode south east towards the border. Just after Kosh-Agach we entered the Border Zone where you must stick to the Federal road, the M52.
We were stopped at a border guards temporary spot check just south of Kosh-Agash, exactly where we wanted to turn left to see if the track to Murgur-Aksi was viable. We bluffed that we were going to Mongolia (Colebatch had a Visa for there and pushed his Passport forward - I did not, so kept quiet!!). We went a short way down the road, and once out of sight and sound, then cut left overland to pick up the track later.
A very hard day and a half later, at Murgur-Aksi, while looking for fuel, we were stopped by some off-duty border guards asking what we were doing there. We diverted the conversation to fuel availability for our bikes. The fuel garage was dry, but the same guards sold us some petrol they had in a container in their truck. They then directed us to their border garrison, right on the border further south-east, to get permission to continue along the border track towards the Chadan road.
The Commandant was friendly, gave us tea and eventually his verbal consent to continue - as long as we stayed on the track. Nothing in writing.
The journey was then uneventful except Colebatch had a flat tyre that caused us to camp the night within sight of the garrison. The Commandant tooted and waved to us from his vehicle as he went off somewhere for the night and again when returning next morning.