Bluskis
Yes, the airmass is very relevant. If you take my example of the Azores High. The air starts off in sub tropical latitudes and is dragged northwards by the anticyclonic flow. As the air moves northwards over an ever cooling ocean, the airmass also cools. Therfore becoming more stable. When the airmass reaches the UK, it is still warm relative to the airmass that is being replaced, but cooler and therefore more stable than when it started its journey.
On the flip side a cold, dry and stable airmass originating from the arctic increases its moisture content and increases in temperature as a typical low pressure sitting over Iceland drags the airmass south and southeastwards. As the air moves southward over an ever warming ocean it produces even more instability and therefore deepens the low.
As with anything in Meteorology you have to understand a number of basic theories before you can understand why we get any particular type of weather. In this case you need to understand, airmasses, stability/instability and as was eluded to in the original question, the relationship between pressure, temperature and density.