This is known as transfer function model:
For stabity and invertibility you have to have some restrictions on the characteristic polynomial of this representation. Which means processes jointly has to satisfy these conditions.
You can have seasonal event dummies, intervention dummies or other deterministic components here without causing too much trouble. Problem is to identify them on the first place!
You can think ARIMAX model as a stochastic difference equation which can have Dirac delta function type of events with their own constant coefficients.