well... generally there isnt a reason for used tyres to have less grip... slicks have rubber on rubber, so up to the point when you are rolling in the structural mesh, you are always driving on the same rubber, so you should have the same grip (for the same temp). For treaded tires its different - the more worn, the more grip, cause it begins to resemble a slick more and more.
In road cars, since when the tire is reaching its "slick" phase and needs to be changed (unless you are 100% positive it wont rain, nor will you find any pebbles/sand/small stones on the road), you SHOULD have more grip.. but you don't. This is cause by the time the tyre reaches its slick phase, its 2-3 years old (lets say 40'000kms), and the sun has cooked it till its much harder than before. It is also much more brittle - you can see this by looking at the sidewalls of an old tire - between the sidewalls and the treads, you can see small "cracks" in the rubber - it has hardened, and provides a lot less grip.
It IS normal to have faster laptimes at the end of stints- even in RL - just look at f1's times (the easiest to find laptimes) - they do good laps all during the sting, and the times usually shrink (due to a lighter car as it burns off fuel). Only on long stints the last laps aint that fast, cause a) driver is conserving whatever left is of his tires b) driver is nervous his tire might not last and propell him headfirst into that hard hard wall at 200kph and c) tires are very cold, cant heat up, and provide less grip. Usually all3.
Takes a special driver to really take advantage of the cold end-of-life tires and the light fuel load and make some mind-boggling fast laps (schumi/senna etc) - one that knows just how much the tire will still last from the feel it gives - from the temperature its at (taken from how much grip its giving), and for how hard its been pushed, then estimate if it will last just one more lap or not