when the strata is saturated with water then the occurrence of sinkhole is rare or should not happen at all. Consequently, as much as the surface soil is dried as much the sinkhole will take place. we can say when the soil is dry enough is birth of sinkhole and as soon as the soil started to store the water in it, the death of sinkhole will begin.  

Of course there is one condition, this condition is the availability the existing of karst cave laying under the surface clayey soil.

I'm not aware of a formalised theory/framework of this.

It's going to depend on the type of sinkhole as well. For a suffusion sinkhole (for example) then it is more progressive, rather than having a fixed end point, so it's hard to conceptualise a "death". Whereas for a cover collapse type sinkhole in karst terrain, then it could be conceptualised as the start of the dissolution of limestone is the "birth" of the sinkhole or the start of the overall process. Then once the void is sufficiently large to create a void into which the caprock collapses this is the "death", as it is the end of the active process. It tends to be the case that caprock type sinkholes aren't progressive. That is to say, once the true sinkhole has formed the feature is no longer "active" and it is rare for them to keep expanding.

There certainly isn't a conceptual theory of this though, in text books for example. It's an interesting angle to ponder.