Take an interval. Cut it into H pieces, where H is hyperfinite. This serves as the index set of a stochastic process, among many other uses. Imagine that for each of the H steps, you flip a coin to get -1 or +1. Then move an infinitesimal distance left or right based on the sign. This is Brownian motion. Each infinitesimal piece of the timeline is profitably thought of as a Planck time.
Discrete events, such as sudden hard shocks, can be modeled on this line. They are appreciable over an infinitesimal fraction of the line.