If you download it after you're off of that type of case then do you just go back and select the DLC case and it plays as if you're at that point in the game?  The reason I ask is because I have not yet finished LA Noire, but can tell I am very near to the end.  My next case is Nicholson Electroplating Disaster.  I know this is a DLC and the beginning of it picks up right where my last case left off.  So for the people that download them before playing the game, do they just stick them in the correct time periods for you and it plays smoothly, but for the people that downloaded them after, you just have to go back and play them?

So for the people that download them before playing the game, do they just stick them in the correct time periods for you and it plays smoothly, but for the people that downloaded them after, you just have to go back and play them?

They are indeed added to the story at the relevant points without any extra input from the player. 
 
If you're past their appropriate slot you have to access them via the Cases menu, or start a new game.