IMO, the "continuity" (if there ever was one), was OS-DP. Ash felt like a true evolving character.
BW broke it apart with the personality and skill 'resets'. It tried to connect back to OS-DP with some continuity references, viz. Dawn and Charizard, but it didn't really fit.
XY felt very self-contained. Ash returned to form and was pretty mature for the most part. However, the references to previous sagas was minimal. Morever, no reserves showed up in the Kalos League, which made no sense as Ash deliberately explained why he wasn't using reserves for the Hoenn League, but had no problem using them for the subsequent Battle Frontier and Sinnoh League. Obviously Ash was skilled, but he was not necessarily carrying over the Countershield/match-up strategizing that he did in Sinnoh. If anything, BW-XY felt like its own thing.
SM turned Ash into a concept. Mayybbbeeee you could say that he was vacationing and chilling out on the beach after Kalos, but it felt like Ash was becoming more of a template that could be copied-pasted to whatever particular series tone. However, there were continuity references again. Lastly, I'd say JN Ash probably is just a continuation of SM Ash.
So...
OS-DP Ash
BW-XY Ash -(?)-> SM-JN Ash