The Time Influx by Katie Mitsui

The snag in the fabric of time that engineer Johnathan Davidson's time machine created dragged every person in in the city around him out of the present, and each reappeared at random over the course of the next 100 years. Though it irreparably confused their lives and relationships and caused an entire network of professional and scientific support to be built around the time survivors, the close of the wormhole at the end of a century seemed to be the ending of the time disaster. A new chapter in the saga begins when the time machine is stolen by a mysterious entity before it can be destroyed. Terrorist threats to use the machine in populated cities and the discovery of dangerous rogue time zones all over the world necessitate the establishment of the "Watchdogs", a group intent on finding the culprits and restoring order.