The following suffices for our purposes:
(Assumption) Particles can be destroyed: if a particle gets destroyed, its trajectory is discontinued in every inertial frame. That is, if tK denotes the time of destruction relative to inertial frame K, then the trajectory will not exist for any t > tK in K.
Note: backed by the fact that the speed of light is finite, similar can be assumed about signals too.
Universality alone would only guarantee that the event of destruction happens in every inertial frame, at exactly one (x, y, z, t) tuple. However, it states nothing about the existence of the particle before or after t. That's what causality tells us in addition.
(Theorem) The order of two events e1 and e2 that happen directly to a given particle is the same relative to every inertial frame.
(Proof) Let K and K' be two inertial frames, and let e1 and e2 happen at t1 and t2 in K, respectively, with t1 ≤ t2. Let t1' and t2' denote the corresponding time values in K'. Moreover, let the particle get destroyed in K at t2, which corresponds to t2' in K'. Then, since the particle does exist in K' at t1', t1' ≤ t2' must hold. From universality, t1' = t2' if and only if t1 = t2. Thus, t1' < t2' if and only if t1 < t2.
The argument can be applied to signals too:
(Theorem) The order in which a signal meets two given entities is the same relative to every inertial frame.