Does near expiration date milk continue to "age" in a cooked dish?
You have either a "Best before" or a "Use by" date.  


If it's an unopened pack with a "Best before" date, the milk should still be perfectly safe even quite some time after that date (it's sterilised or UHT treated) in the unopened package.  
If it's an unopened pack with a "Use by" date, it depends on how the pack was stored: a "Use by" date is for raw or pasteurised milk, which should always be kept refrigerated. And normally, the milk should be used on or before that date. 


If the pack has been opened, you should be able to keep it for a few days at 4°C (refrigerator), but not beyond a "Use by" date.  

I use as a rule of thumb "don't use it if it smells off". If the milk still smells good, when using it in a casserole that is baked, the leftovers should keep normally. In a cheesecake mix, if the milk isn't reheated, I'd be more careful, and not go beyond a "Use by date", or 3-4 days after opening the package.

And the "going off" isn't a matter of aging, but of microbial growth. If it's lactic bacteria, you'll finish with yoghurt; if it's something else, all bets are off...