Does relativity happen at the source ?
Relativity applies everywhere. However, something you seem to be missing is that "relativity" is a comparison of the how an event or events, unfold when observed from two or more different interial frames of reference.
The Lorentz transformations, which lie at the heart of special relativity, provide a mechanism through which observes in different inertial frames of reference can convert what they observe to what an observer in the other frame of reference would observe.
There is a second role that the transformations play. In the second situation the Lorentz transformations can be used to calculate or determine how time is dilated relative to an object's velocity. This affect, time dilation, is unobservable under most common every day velocities and only comes into play when velocities of objects with mass, become relativistic.., reach a significant fraction of the speed of light.
An object, which has mass when accelerated to a significant fraction of the speed of light experiences time dilation. The progress of time for such an object slows down as its velocity increases toward "c".