Does the time that the Avenger spends traveling affect any countdowns?

It appears that the internal timers are identical to the date/time representation shown on the geoscape: if time's passing on that, the timers are progressing.

However, not all timers are actively checked if the Avenger is airborne: project-completion and site-expiry events seem to only trigger while the game is on the Geoscape screen with the Avenger landed. Even if the timer for a project or mission site hits zero while the Avenger is in flight to a new location, nothing actually happens until after it has landed at the destination. Since landing at a mission site jumps you straight to the start-mission screen, this means that once you give the 'fly here' order for one things that would normally trigger in-flight are queued up until the next Geoscape visit after the mission ends (or until the player returns to the geoscape using a 'cancel deployment' option, which means such options should be used with care when dealing with time-sensitive missions).