I am not sure you are getting this. A company has say 4 AN124s, say, for the sake of argument, each one gives 1000 hrs a year in flight time, so 4000 hrs a year. Company based in Europe.
A country then says okay I am having an exercise/deployment in June coming back in August I need 100 hrs of flight time. So you pay the company for 100hrs of flight time. There are other variation of this, e.g. you can book time per year and on sell/trade it etc...
Does not replace existing assets. The NZDF has used this in the past and will no doubt use it again.
Where it falls apart is in the event of an emergency like Tsunami, ET etc..because you can't always have it available (someone else may be using it)
Hence some nations have looked at buying the airlift assets in a civilian config and leasing them to commercial interests, booking their time for regular military lifts, but with emergency clauses in the contracts that say we will use them in case of a disaster, war, emergency deployment etc and reimburse for lost earnings. Or they are just buying their own Strategic Lifters.
For something similar look at what the RAF is doing with its AAR.
FYI the ADF did use the AN124 (or II-76) for the initial deployment to ET in '99, and for all I know may be using it again.