quantum mechanics - Is an elementary particle traveling through a vacuum the *same* particle at points A and B?

This is a question I've wondered about for a long time.

Imagine an elementary particle moving through a vacuum. Take any two arbitrary points along its path; we'll call them points A and B. Is the particle at point A the same particle which is later at point B? Or is the thing we perceive as moving through space more akin to a wave moving through water (where the "energy" moves but the water molecules comparatively little)?

As I understand it:

1. Space has a resolution to it (the Planck scale?) and elementary particles are excitements of fields


2. Elementary particles have no definite location (dancing somewhere between being in multiple places at once and popping in an out of existence)

The first part implies to me that space could be thought of a bit like a LCD screen (with pixels) and each field would be one of the colour (RGB) components of each pixel, which extends across the entire screen. In that case, particles moving through space would be more like pixels lighting up and turning off, giving the illusion of motion. The second part implies to me that notions like location and the long-term existence of particles may be a misconception altogether.

Sub-question: if individual particles from one moment to the next can't be said to be the same particles, does that pose ship of Theseus questions for macroscopic objects? (E.G. a rock floating in space.)