Nice thread, I must say
perhaps I can add a word or two which can't do any harm but can help in some situations. I never drove though India, much less over Himalayan mountains. But in Serbia, where I am from, we also have muddy mountain "roads", loose gravel surfaces, landslides etc.
You don't have to go high in the Himalayan to fall off some 50 meters below during landslide. You only have to go 50 meters above something and then fall down
I don't know if you have wheel chains or snow chains in your vehicles, but those can help in mud and slush as well as on snow. Simply, you will have little bit more grip on your wheels.
When I go to a mountain, which is almost every two weeks, I rely on equipment that I always have in my Safari. Never go to off road without tow rope. Any rope, strong enough, is far better than no rope. But the best is if you have synthetic tow stripe or tow braid instead of steel tow rope. Those look like fire hose. Much lighter and stronger than 10mm thick steel rope.
Tow braids can be found in two versions: static and dynamic. Static does not stretch, dynamic stretch itself for certain amount. I like dynamic for getting me or somebody else from mud pit because by stretching it gives additional momentum and the risk of ripping a portion of car body is reduced.
Shovel, spade, pickaxe, axe and mallet are something that I find very useful on terrain. And those are not pocket size, easy to carry, folding, multipurpose, toys. Normal shovel and spade with normal wooden handles. You don't need user manual for those tools
If you shove floor maths under wheels, please tie them with longer rope to a car body. If you manage to get yourself out of mud and slush, maybe you don't want to stop and pick the maths. This way maths will follow you to the first secure place where you will stop and pick them.
Be careful when choosing branches and logs underneath the car. Pick larger number of small and soft branches rather than one or two big and hard logs. Try to put them in a way that they will not hurt underside of a car. Branches have notorious habit to jump and bump to a car mostly on most expensive and important parts
(the Murphy's law)
All of this will make you less dependent of mercy of others. And yes, tell your passengers to stretch their legs and to go before you but not far from the car in case they have to stand behind and push you. Or leave them inside if you need their intelligence measured in kilograms to give you better grip