At the beginning of a sprint, a project manager or Scrum Master moves tasks off the team's backlog project into the new upcoming sprint. In Phabricator you have to edit each task in turn, removing one project and adding the other.

1. Drag and drop would be perfect for this, except you can't drag tasks between two Phabricator windows (FWIW you can't drag cards this between two Trello board windows either).
2. Trello has a Move action on a card (task) where you can choose a different board and a column within that. Phabricator can't easily do that because a task can be in multiple projects, but maybe there could be a UX to get to this by interacting with one of the task's Projects tags.
3. In Trello from a board column's (▼) drop-down you can choose Move List... and Move All Cards in This List... This lets you organize tasks into a "Ready for next sprint" column and then do a bulk move.

3 seems like a straightforward enhancement to Phabricator workboards which have a similar ▼dropdown on each column. I think the dialog could just let you choose a project or sprint to move to. It might be nice to be able to choose a position for Move Column... and a particular column name for Move All Cards in this Column... in the destination board, but the defaults (new first column and the default column) are usually what you want.