S/O upgrades to F/O:
They will upgrade you wherever they need crew.
You will start as an A340 S/O, then after a year or 2 do a CCQ to the A330. Some guys upgrade from airbus to 777 F/Os, some will stay on Airbus. If you upgrade from an Airbus S/O, you will initially only be an F/O on the A330, later on, they may CCQ you to A340 F/O. Where they can, they generally upgrade Airbus S/Os to Airbus F/Os as it is a shorter course and therefore cheaper for them.
777 and 744 are completely separate type ratings, having said that, 744 and 777 are very similar to operate in terms of the logic of the AFDS and FMS/CDU, so it is a lot simpler to jump from a 744 to a 777 than to an airbus.
Even when you are an FO they will switch you between airbus and boeing if they need crew in different fleets.