To the best of my knowledge, all the Russian attempts have been with diving bell-type rescue vehicles hanging down from a surface ship and these sort of things get swept away by the current. I'm quite hopeful that it can do that. The LR5 has only three crew members, of which two are pilots. That leaves just one man to enter the Kursk. Its normal operating crew is one pilot and one man in the rescue chamber but for the first mission down, they would probably take two pilots so that they can both get familiarised with the layout of the distressed submarine.
Reports talk about the gradual depletion of oxygen as limiting the crews' survival prospects.
Whereas some have said that the build-up of carbon dioxide is in fact the limiting factor for the crews' survival. The carbon dioxide problem is exacerbated by pressure so the higher the pressure within the submarine the more dangerous that becomes. But I'd like to think the Russians, like all well-trained submariners, will have minimised such problems by slowing down their breathing rates so they consume less oxygen and exhale less carbon dioxide. The problem here is they'll be getting cold and when the body is cold it is inclined to shiver to warm you up and when you shiver your heart rate and therefore your breathing rate increases and you use more air. It's a matter of trying to keep yourself as placid and quiet as possible.