The flexibility of General Relativity lets one consider the same physical picture in different ways, which can aid in understanding. It can also serve to confuse too! :-) A fair amount of research in General Relativity is in laying down some rules which lets one reject a mathematically consistent and complete picture as unphysical, for example, or alternatively which lets a picture that is only partial -- or which can't be proven to be complete or wholly consistent -- as a good physical approximation.

> Travelling along a curved spacetime wouldn't imply a force on a neutral particle, would it?

There are exact solutions of the Einstein Field Equations (although typically this is done with the linearized version of them, the LEFEs, which is in the "almost certainly a good physical approximation" bucket) in which a change in the background metric propagates through vacuum in some direction as a plane wave.

The background metric could in principle be, for example, flat spacetime, or Schwarzschild, or any other vacuum solution of the EFEs. When one chooses a splitting of spacetime into space and time, the wave propagates through space.

A small set of test particles reacts to the wave as it passes over them, changing their respective "radar distances" for example. This view lends itself well to perturbation theory, where the wavelike perturbation of the background metric can (if weak) be quantized; those quanta are gravitons in perturbatively quantized gravity. That's a real theory that exists and is good everywhere where there is no strong gravity. "Strong gravity" can even be defined in terms of perturbatively quantized gravity as at least one loop of gravitons on a Feynman diagram. You generally don't find that until very close to a gravitational singularity (i.e., perturbatively quantized gravity as an effective field theory works inside stellar BH horizons too, and works even better inside even more massive BH horizons because there you are further away from the strong gravity near the centre.

In the spacetime view, the particles are instead worldlines that are mostly in regions described exactly by the background metric, but a small section of the worldlines are found in a small region where the metric is different. One can consider the spacetime intervals of points on the various worldlines and look for some minimum which will ought to be found in the region with that different metric. Exactly oppositely from Euclidean geometry, curved lines are shorter than straight lines, so a(n inertial) worldline which is found only in flat spacetime regions will be longer than one which is (even briefly) also found in a region with the slightly different metric.

Are such worldines travelling along? Nothing really travels at all if it's a worldline in a "block universe" spacetime picture like in the paragraph immediately above. But if one foliates spacetime into spaces with a time coordinate, movement emerges. You can choose a foliation and set of coordinates in which the different metric's effects are shoved into time-like effects (if the background is flat spacetime, a particle exposed to the gravitational wave is younger than a particle that isn't; c.f. the twin paradox); or you can choose a foliation and set of coordinates in which the different metric's effects are shoved into space-like ones ("radar" showing that the particle exposed to the gravitational wave is dragged or bobbed by its passage).

With some of these equivalent and (mostly) convertible views of the same physical picture, it would be perfectly reasonable to say that a force has been exerted. After all, one is interested in what causes a small piece of a long stable worldline to look different. The force acting on that section of worldline (or on the particle during that brief period of time), however, depends on one's choices of foliation, coordinates, and a few other things. If you can make some set of such choices such that the force vanishes, we call it an inertial force, d'Alembert force, or frame-dependent/fictitious force. Real forces cannot be made to vanish by any choice of coordinates etc.; real forces' "fingerprints" must do more than change an object's position or orientation in spacetime.

The tendency of objects -- particles, worldines, extended (many-particle) objects, worldtubes, or the full fields in which these worldtubes are just clusters of correlated numers -- to look mostly the same is one way of thinking about inertia: you'd want to attribute any unusual change to some force acting at the point of the change.

If you remove everything but your test object, so that no other object or field can impart any non-d'Alembertian force on it, then a magic observer at an enormous distance watching its behaviour could say that any changes in radar distance or (if the test object is a clock) ticking rate is the result of coupling with curved spacetime. But a magic observer who can see the whole worldline of the object and knows the layout of spacetime curvature will note that it matches a longest-possible solution of the geodesic equations.

> I don't want to burden anyone with my questions.

It's not a burden!