I've been meaning to write a post for a bit now about my thoughts on the circumstances of Dess's disappearance, so here we go!
Based upon the sparse info we've been given in-game, I believe only Kris and Dess were present for the incident. While I used to believe Noelle and Asriel were also witnesses, Noelle's remark about them "never finding anything in the woods" killed the idea. While, yes, Noelle is known to block out traumatic memories, she seems genuinely perplexed as to why Kris became distant following the incident. In chapter 2, we also learn Noelle has a tendency to search Dess's name online, which wouldn't make much sense had she, for example, watched Dess disappear into the Shelter.
I also believe going to the Shelter was Dess's idea. The code within the guitar is pretty damning evidence of this; I doubt Carol would've put it there given how preservation-centered she is. So it's likely Dess acquired all of the codes, with her mother's ending up within her guitar. As for the other two, we may find them hidden in other places related to Dess.
Additionally, I believe Kris was indirectly responsible for Dess's fate. If I had to guess, they possibly pulled a prank that hurt her, and this triggered a series of events leading to her becoming shattered across time and space. Or, alternatively, Kris opened a dark world in the Shelter in hopes of showing Dess how awesome they are. However something about the Shelter's dark world was wrong, and it somehow consumed Dess before she could escape with Kris.
Lastly, I believe Asgore's involvement in the actual incident was minimal, if not non-existent. I severely doubt he knows about dark worlds; if he did, he would've created one in an attempt to prove his sanity and save his marriage by now. Instead, I think he somehow screwed up the subsequent investigation, something which he would've been directly involved in as police chief. Like in Undertale, he likely did something quite cowardly to avoid causing further distress. As for the specifics, I expect we'll learn the full story in chapter 5.