Reference Manual > Editing > Common Editing Actions: "Audio Event Slip"
Often, after an Audio Event has been sized to fit a particular region of time, the audio clip the Event contains needs to be moved ahead or behind in time without changing the Event’s length and volume envelope. This action is commonly called “slipping” or “slip,” and it is often used alongside splitting, or splicing, to correct the timing of rhythm tracks. For instance, if one snare-drum hit is off the beat by a little bit, you could split the Event on either side of that section and then slip the audio into perfect time.
To use Slip, select the Arrow tool, and then press and hold [Ctrl]/[Cmd]+[Alt] on the keyboard, while floating the mouse over an Audio Event. The Slip tool icon appears. Click-and-drag on the Event to slip the audio left or right across the timeline. Multiple Audio Events can be selected and slipped at once, even across multiple Tracks.
When slipping the audio in an Audio Event, note that all of the Event characteristics remain unchanged, including the Event size, position, Inspector parameters, and volume envelope.
An Audio Event can be slipped only as far as the length of the audio clip it contains
It doesn't appear to snap to the grid currently.
Check the feature request board to see if that feature has been requested or not and if not, create a feature request for slipped audio to snap (perhaps) it's transients to to grid.