Our story begins in the near-future as our present. A catastrophe hits the entire world which results in time itself collapsing. Nobody in the present day knows what happened or why, but the problem is rapidly getting worse. The world itself seems stable, but soon enough, people are being pulled in from other time periods into the present. People end up with versions of themselves from different points of time all existing in the same moment. For adults, they usually have one much younger, one plus or minus five years, and one about ten years older. They're not all from the same years but they're generally close to that model. You could have some interesting moments where people realize something bad is going to happen to them because they don’t have a future version. Or some questioning teen meets his future self and she’s a lady, leading to some soul searching and acceptance happening earlier than it normally would have.
Society initially struggles with the massive population boom, but the people from the future have knowledge and new tech to share. Automation grows rapidly and with the state of the world as it is, fewer people have kids so there's less threat from new population. As things calm down, people start to realize that there's nobody from the future that’s past a specific date. The latest people that appeared were all from right up to that date but never past it. People stop worrying about the issues society has faced so far and research facilities start going all in on collecting information from the most-future group of people and trying to figure out that time’s significance.
As time goes forward without answers, people devolve into panic and society starts to shut down as we approach the date. Some people see it as the end of the world and for some, the date is when the incident that caused time to collapse is going to occur. People turn on the research institutes and scientists as if it was a science/government experiment gone wrong that caused all this.
I really only just had that setup, but I liked it. No real conclusion or answers, and I think the real story would have been similar. Maybe everybody’s panic about preventing the date causes a societal collapse before the date even arrives. Maybe the study of the event is what caused the event to happen in the first place. Maybe the event will just keep happening and resetting to the initial present over and over it each time it happens, there are more and more people from the future being sent back causing bigger problems each loop. Or maybe we just follow a normal cluster of a person and they hang out with their past and future selves, becoming better people and enjoying the end, and since they’re not scientists themselves, they don’t discover any amazing answers and just get rumors like everybody else.