How was Sazh in 4XXAF? Did he go with Hope and Alyssa? Even if that is the case how come his son never aged? By the time Hope develops the technology to makeshift time travel Dajh should be a teenager. It feels as if they threw him in the end of the game for kicks without a lot of thought behind it.

@ImmortalSaiyan: I had thought of the same thing but then again if you REALLY start to analyze a time travel story you'll make your head explode. Best to just go along for the ride. Final Fantasy 8's story was a mess as well because of all the time travel nonsense. One of my favorite in the series mind you... but still a mess story wise. To answer the question they probally did arrive in the future like hope and Alyssa however like you said the Dajh thing makes no sense.

I assume you mean 500 AF, correct? I don't remember him in 4XX. 
 
Also it's just what he has been doing behind the scenes. If you do more sidequests (specifically the stuff in Vile Peaks) you will learn that he had also been time traveling but wasn't sure how or why. It's probably safe to say given the age of Dajh there whatever happened to those 2 to make them time travel happened very shortly after the events of FF13. If you ask about Sazh in the live trigger when you first meet Hope he mentions they don't really know where Sazh is other than going away to fly airships. 
 
It's all there but you're going to have to do sidequests to learn more about him and his situation.