the hydrocarbon portion is highly volatile, especially in really cheap lithium spray grease (the very low viscosity stuff), so it can actually dry up completely into just white lithium soap. because of this, the white lithium that eg honda vends for greasing joints in their cars and as a dieletric uses a thick silicone base iirc and is really viscous. for anything that actually needs to be lubricated, they point you towards their high temperature urea grease, which is also highly viscous, but doesn't degrade and is cheap enough to fill rubber boots with.
now that i've had enough experience with the various non-exotic krytox formulations, my feeling is that polymer oils in a ptfe matrix are the only way to go for small parts and in particular for linear motion. and i mean for everything except for ultra high temperature applications that will degrade the polymers or for situations like ball joints where you the joint is jacketed, needs to be packed, and needs to be refilled fairly often.
that said, there are a lot of terrible polymer lubricants out there. for small parts, my feeling is PFPE or bust. since dupont is the only one making a full line of PFPE lubes at the moment, that's what we buy here, on the korean forums, etc. superlube is often vended in place of krytox where there's price sensitivity, or solid ptfe is used, but solid ptfe doesn't have the abrasion resistance and lubridicity alone that pfpe gives it, and superlube is, excuse my french, le merde.
as for other bases, silicone is best used as a dieletric and elastomer component, and most hydrocarbons damage pretty much everything, in addition to being ridiculously volatile.