Let's start with dragoon, since it's the job that symbolizes Heavensward. Most of the change to dragoon is contained within the new "Blood of the Dragon" action. Upon using this ability, action strength is increased for a short time. With the effect active, you can use two new abilities to lengthen the Full Thrust and Chaos Thrust combos. More importantly, the effect duration of Blood of the Dragon can be prolonged in this manner.
The strongest of the abilities that can be used with Blood of the Dragon active is Geirskogul, which is a frontal AoE attack learned at level 60. This attack has the additional effect of shortening the duration of Blood of the Dragon. Since Blood of the Dragon does not have a very long initial duration, if you use Geirskogul just after activating it, the effect would end immediately after. However, you can first prolong the duration with the previously mentioned actions until you have enough remaining duration so that it does not immediately end.
For dragoons fighting at level 60, by properly using Blood of the Dragon and timing other buffs to be used when the effect ends, DPS should be enhanced, so it will be like two phases to the DPS.
After using the Blood of the Dragon, if you don't do anything to extend its duration, it will end quite quickly. You can do things like first accumulate the Full Thrust or Chaos Thrust combos to a certain point before using Blood of the Dragon, so taking all of these different things into consideration should noticeably change the course of battle.