All else being equal, longer chainstays equate to a longer wheelbase, and the rear wheel trailing further behind you. You are correct in your assertion that as wheel size increases so too does chainstay length, though many manufacturers have put considerable effort into minimizing this and as such the increased chainstay length is not directly proportional to increased wheel radius. For example, a 26" and 29" version of the same bike model are unlikely to have an inch and a half discrepancy in chainstay length. That said, there are tradeoffs to be had in chainstay length differences.
As an analogy, think of a drag strip or hill climbing motorcycle vs a standard sport or dirt bike. The first two examples have very elongated swingarms that put the rear tire well behind the rider and dramatically increase the wheelbase. Those long swingarms are great for preventing the bike from looping out and maintaining straight line stability, but neither can turn or corner like the latter standard motorcycle examples. That is a very exaggerated example, but the same principles apply to chainstay length differences on bicycles. Again, all else being equal, longer chainstays are going to be more stable at speed and typically provide better climbing traction, but they will cause the bike to not handle as sharply and to no turn as quickly- that much is just basic physics.
Keep in mind that chainstay length and wheelbase are just two factors in a multitude that determine how a bike will handle. Also keep in mind that just because you have a bike with slightly longer chainstays doesn't mean you can't rail corners or navigate switchbacks, and just because you have shorter chainstays doesn't mean you can't bomb descents- much of this comes down to technique, and you'll alter your technique to accommodate for any differences in your bike's handling characteristics.