The many-worlds interpretation and a free particle How does the many-worlds interpretation understand the time evolution and spread of a free particle wave function? That is, does every continuously small change constitute a cosmic action that causes the whole universe to split into multiple realities? (As you may notice, I'm not a big fan of this interpretation.)
 A: 
How does the many-worlds interpretation understand the time evolution and spread of a free particle wave function? That is, does every continuously small change constitute a cosmic action that causes the whole universe to split into multiple realities? (As you may notice, I'm not a big fan of this interpretation.)

The many worlds interpretation (MWI) claims that the equations of motion of quantum mechanics are an accurate account of how reality actually works. I'm going to explain the MWI and then give an answer to your question. When a quantum system $S$ with the state $\sum_a |a\rangle_S$ is measured by a measurer $M$ in a ready state $|0\rangle_M$ we get the state:
$$\sum_a |a\rangle_S|a\rangle_M$$.
The information in the measurement device usually spreads to other systems called the environment and this spread of information prevents interference between different measurement results: this process is called decoherence. The resulting state looks approximately like a collection of parallel universes, hence the name of the MWI:
https://arxiv.org/abs/1111.2189
The measurement results spread by local interactions so a system won't see multiple versions of the measured system until it interacts with the measurement results:
https://arxiv.org/abs/2008.02328
So a measurement doesn't produce multiple versions of the entire universe.
If you have a free particle propagating without interacting with anything else, then instances of the particle in different positions will still be able to interfere with one another and the parallel universe approximation won't work. If you measure the particle then the measurement results will decohere unless you take elaborate precautions to prevent the spread of information about the measurement results and then you will have a collection of parallel universes. The presence or absence of interference between different states is often a matter of degree so counting the branches is a bit tricky, but interference for an object as small as a grain of dust is usually negligible on any scale you can so quickly that you won't see it, see pp. 12-14 of
https://arxiv.org/abs/quant-ph/0306072
