In your question you have a photograph of tracks in some kind of bubble chamber. This is an old technology, in which some kind of fluid is prepared in a supercritical chemical state, on the wrong side of a phase change. (In a "cloud chamber" the fluid is ready to condense liquid from vapor; in a "bubble chamber" the fluid is ready to boil.) When energetic charged particles pass through such a chamber, they create ions along their path of travel (hence the name "ionizing radiation"). Each ion serves as a nucleation site for the supercritical phase change, so the ionizing radiation leaves a trail of droplets or bubbles which are visible to the naked eye. So the procedure for using a bubble chamber is something like
- Prepare your chemical's supercritical state (e.g. by pulling a vacuum on a liquid
- Send a pulse of ionizing radiation through the chamber
- Use some side-channel information about the pulse timing to trigger the shutter on your camera
- Get the supercritical fluid and the camera ready for the next pulse.
On the time scale of a chemical phase change, the motion of any relativistic particle across a fluid-detector volume is instantaneous. You do get motion in a bubble chamber: the bubbles move. (In a cloud chamber, the droplets fall.)
The chemistry part of this is slow, and messy, and expensive. Physicists don't like using chemistry for detectors. We like to use electronics.
In a modern gas ionization detector, we don't detect the ionization by waiting around for a chemical change. Instead we fill the volume with wires, run the chamber at high voltage, and the ions fall into our data acquisition system.
If the wires are arranged in planes mostly perpendicular to the direction of the incident radiation beam, and the wires in one plane are mostly perpendicular to the wires in adjacent planes, then we can reconstruct a ton of information in both space and time. Perhaps one event has the most charge collected on wire 42 of plane A, on wire 86 of plane B, on wire 167 of plane C, and on wire 230 of plane D; this is already a track, and there are clever tricks you can use to get a position resolution that's much finer than the size of the gaps in the detector.
(You can also collect the ionization released with your radiation passes through a semiconductor, like silicon or diamond. These detectors are smaller and so have better position sensitivity; they are faster and so have better time sensitivity. They don't last as long in a high-radiation environment, though, and replacing the silicon in a wafer is more invasive than purging the gas in a wire chamber.)
The Wikipedia page on time projection chambers, which is one of several ways to get timing information from a wire chamber, is exactly as not-helpful as the first ten hours of seminars I sat through on the topic — it's a niche subject. But the short story is that, in a modern particle detector, timing information is absolutely part of track reconstruction. And in answer to your title question, I think probably 60% of physics graduate students who get a pile of position-and-time tracking data will make an animation of it. I've seen a dozen such animations (and made two), but only one (not mine) actually solved a problem and showed up in talks and theses.
In response to your extended question: the detritus of a collision always starts at the site of the collision and travels outwards, forwards in time. In collisions which produce antiparticles, it is not the case that the antiparticle detritus appears before the collision and travels towards it. Among other problems, having the detritus reach the collision at the right instant is an impossibly low-entropy requirement; you'd have better luck unscrambling an egg. Feynman's comment is a low-jargon way to claim CPT symmetry. The transformation CP reverses a particle's charge and parity, which makes it mathematically identical to an antiparticle; the transformation $T$ changes the signs of all the time intervals. It is a deep and important result that CPT symmetry is closely related to the way spacetime transforms in special relativity.