A similar thing happens in regard to heading (navigation) on the globe. If you line up with a heading on a latitude line corresponding with your heading, you veer off course inevitably and constantly. Demonstrable on google Earth, as seen in
this video. Latitude and heading are not the same thing.
Of course, there
are no vast-cross-equatorial flights that I can find, so... that would be impossible to test anyway. Although many flights from Australia to Brazil work exactly like they would on the Globe, as can be seen with any booking agency.