Normally (excluding nuclear reactions etc.), it is the electromagnetic ("em" from here on) repulsion at the root of all. A push is nothing but electromagnetic repulsion between what is pushing and what is being pushed.
If what is being pushed is firm enough to exert equal and opposite em force, then there is no motion, even though there is pressure from both sides. Otherwise, it gets displaced by the repulsion which becomes velocity. Pressure is nothing but em repulsion per unit area.
Attraction can also cause pressure, but repulsion is a must to realize the pressure. Just like gravity (attraction), causes pressure between your feet and earth, but the pressure is only realized by the repulsion between electrons in your shoes and those in earth where you are standing.
Faster (in parallel) moving fluid/air has lesser chance of exerting that repulsion and hence bernouilli principle.
So, to answer your question, EM causes push/pressure, which causes displacement, which results into velocity.
In one of your comment - "velocity drops and pressure increases" - basically, the opposing force is firm enough to slow down the displacement and so let the pressure increase till the point it matches the source push.
