Scattered Across Time and Space

Objects or people are sent to not one, but many separate places in time at once. This is often done by result of an accident involving or untrained use of time travel stuff, but can occur due to a time-traveling antagonist hiding objects at various exotic areas (reasons they aren't still there in the present optional) that the time-travelling heroes must locate. This is naturally an excuse for a work to travel to wherever and more importantly whenever the creators deem interesting enough to set some of the work, as the protagonists themselves must travel to those places to retrieve the out of time objects.

Since it involves multiple objects to find and the continued existence of these objects in the past is often hazardous due to risk of Temporal Paradox or leaving a villain free to meddle with history, it naturally pairs with Gotta Catch Them All. If an intelligent being is stuck in the past from this, they might Write Back to the Future so their allies know when to find them.

Anime and Manga

 * Having a wish granted in Dragon Ball involves gathering the seven titular balls together. Once a wish has been granted, the dragon balls are scattered again, invoking this trope.

Fan Works

 * In the RWBY fanfic Emergence, the members of Team RWBY arrive on Earth over the course of several weeks, in locations all over the globe, despite all being affected by the same event. Averted with Team JNPR, who all arrive together in Washington, DC as the result of a different incident a year or so later; Cinder, Mercury and Emerald, transported in the same incident, also stay together, but arrive at a different location and time from JNPR.

Live-Action TV

 * Season 2 of DC's Legends of Tomorrow opens with the cast having been stranded in various time periods ranging from the Cretaceous onward due to use of the emergency Time Scatter functionality.

Podcasts

 * This will happen to you in Welcome to Nightvale if you touch several plastic flamingoes at once. They happen to be time loop devices. Your consciousness will be split over several selves while you are trapped in a time loop. Don't touch the flamingoes.

Puppet Shows

 * The Adventures of Pug and Zero, a 1997 show, was centered around this premise. Two alien time travelers --"quantum surfers" in their words-- borrow a device from a Teen Genius named Charley that can travel through time and space. This device, the Exploragyzer, doesn't have a control button, however, and sends them to random locations and time periods. From Pug having to learn to be a knight in medieval times to Zero learning movie monsters aren't real or the aliens getting arrested in the Wild West, anything goes about where, or when they will end up.

Video Games

 * Where in Time Is Carmen Sandiego? has Carmen deposit her crew throughout time to steal historical artifacts. Now ACME must track her crew down and recover the stolen objects.
 * TimeSplitters 2 has the TimeSplitters take the time crystals to various points of history to hide them from the protagonists.
 * Recovering people, creatures and objects from the past and present of the Wizarding World that have been fragmented and scattered through the Muggle world was the primary goal and gameplay loop in the now-defunct Augmented Reality Game Harry Potter: Wizards Unite
 * Ape Escape starts with monkeys activating a time machine and being scattered throughout history to rewrite the past and become the dominant species on Earth.

Western Animation

 * Dipper and Mabel in Gravity Falls do this by accident in "The Time Traveler's Pig", accidentally spilling the objects in their pockets across time and space through a sibling squabble. Because they used Blendin's time belt, Blendin' is charged with picking up all the objects through time after the Time Police arrest him.
 * Sylvia and Wander briefly end up like this in Wander over Yonder in "The Waste of Time". Wander accidentally grabs time Orrble juice instead of the regular Orrble juice, causing him and Sylvia to travel through the past, present and future..
 * Princess Gwenevere and the Jewel Riders have to travel via wild magic to other lands, to find first the Crown Jewels and then the Wizard Jewels. One stone even will freeze time.