I promised to post my solution and did not do it, so sorry.
What I wanted to accomplish :
I wanted to take the train from Moscow to Beijing, no stop with my then 79 years old father, I lived in HK and him in France.
I needed a Russian, Mongolian, and Chinese Visa.
I'll only write about the part in HK.
1)et your tickets as some countries might want you to prove that you have the ticket out of the country. I did that trough an agency that provided me with both the flight from switzerland to Moscow and the train ticket to Beijing
2) some countries like to know when you are going to get out and where you are going, and sometime prove it, so that mean that you are better of getting the visas in invers order of the visit so Chinese visa first. I had a multiple business visa for china, so that was done.
3) As I was going in and out without stopping in Mongolia I went for a transit visa. I called the Mongolian Ambassy in Beijing, a Nice guy told me to pop in in Beijing for an overnight, promised me to get the visa next day. I told him I was living in HK, that I would rather avoid spending two days in Beijing just for a visa and when I asked if I could send in my passport via courier he just laughed and told me to come to Beijing and have some fun ...
As I did not have time for fun in Beijing, but was going to have fun very soon in Singapore I got my visa there. Do your homework, download the forms in advance, get the ID pictures, have the money ready, The Mongolian Wrestler at the reception looked like he game straight out of a very heavy boozing night, barely understood what I wanted, was thrown off by the fact that I had all I needed and lost completely track of what was going on because I did not need him to laboriously explain me what he needed, on the other hand if you don't have what is needed on hand you might spend one more day to get your Visa. BTW the Wrestling guy was very nice, just lost in translation. The day after there was a lady as lost as the wrestler, she first asked me what I wanted, was totally surprised that I was there to retrieve my passport, tried to get me to fill in all the papers again, that took 5 minutes to find my passport, I could see the pasport, it was in the drawer she opened 3 times, but she did not seem able to identify the swiss passport for what it was (It's red and has a white cross on it and of course my picture...). She then called a young man that solved the problem in 30 seconds, he did not seem to have spent the night sipping vodka :-). btw, not complaining, it was entertaining and everybody was very polite and nice.
4) The russian visa was no issue inHK, a very pretty young lady was nitting at the reception both time I went there, It seem that her only role is to point visitors at an other lady that was not nitting that seem to run the whole show. Again no complains, all nice people. The surprise was that the Visa for a swiss passport was much more expensive than for other countries, I wonder what neutral Switzerland did to the Russian guys to deserve that ?
Final word, really get your papers in order, while entering and exiting Russia and China is not a big deal, entering and exiting Mongolia took really time, with multiple visits of multiple officials asking to see our documents and making a serious search of the compartment, The police and custom ladies where very nice and actually very nice looking, but I got a clear NO NO message from an Armed guard outside the train when I tried to take a picture of the train station, The guy was massive, I measure 1m87 but the guy was towering me and had twice my shoulders.
Sorry for all the rambling, but I really had a great trip and lot of fun and can't help but share a few stories.