Just an update on our experience pre pandemic. August 2019. We were able to put two bikes in the guard van on the Ullanbaataar to Moscow train, all due to failed water pump, however it caused chaos in Moscow as they stamped our bikes as luggage not transport on the Russian side.This meant they were liable for import tax.
Two days of visiting various customs offices later we were free to collect them with just an admin charge. The customs in Moscow were really good and apologetic that their officer at the border was such an idiot!
However, they had to construct a new procedure to deal with us and only one officer had the authority to sign the paperwork in Moscow. Fortunately his holiday was the previous week so only a two day delay. Still, we got a BBQ and accommodation with one of the officers and an official hat to apologise.

They explained that next time ride or push your bike over the border but don't leave Mongolia with it on the train! Too much paperwork. however, the Mongolian version of the Trans Siberian train is much nicer than the Trans Siberian. Great food and the bikes were roll on roll off.

You can put it on the train in anywhere in Russia and from Moscow to Vlad or Irkutsk may save crushingly boring and dangerous Russian roads. The guys at the warehouse in Moscow speak some English and pack your bike well - we saw some French bikes ready to ship. We would consider this when we head back to finish the BAM road.

Alex at Russmoto is definitely the guy to give advice and helped us greatly with our trip.