If they already have the documentation, visa, passport and air tickets, can they be stopped at the Moscow airport by their own Customs agents and be made to show proof of funds before being allowed to board?
First of all, according to the Constitution of the RF article 27.2 every citizen have the right to leave the RF and return to the RF.
- having secret clearance
- having been drafted
- being a suspect or being accused of a crime
- being sentenced for a crime
- evading obligations imposed by the court
- having procured documents under false pretenses
- working for the KGB
- being declared bankrupt (until the resolution of the bankruptcy case)
This list is inclusive and it is enforced by the passport control checkpoint on exit. (Special restrictions exist on travels of minors to prevent human trafficking.)
Customs has a separate checkpoint, they check for a contraband and such, and may prevent you from flying when something suspicious is found.
Absence of funds is not a valid reason to temporarily limit the right to get out of the RF, and even if it was, it is not checked by the customs.
PS. RF = Russian Federation, full official name of Russia these days.
As @mzu already said, there are a lot of checks on borders in Russia, most of them are being made by passport controls, not the customs. Customs do the checks fir luggage - are there too many cash, are there some items prohibited from moving outside the Russia (such as paintings or rare items), are there drugs, etc.
Neither border control nor customs will check one's financial situation, because it's the point of interest for other country to check incoming people, and sums per day are different in different countries. The most popular case to get some funds for border control is that if one's who want to get outside the Russia has an unpaid bill, like taxes, driver fee, etc. If such sum is over
10000 RUR (about
$200), one could be stopped on border. However, if bill are paid (it can be done tight on border via electronic payments) one could continue theor trip without any problem.
So, basically if you'd have been said that someone need money for going through a border checks, the most probable scenario is that you're a target of scam. Do not send money, try to get back to Russia to see the person and resolve the situation with direct involvement.