Not familiar with LibGDX, but this site may help you.

One of the options is attached, which makes the "particles stay the same distance away from the emitter regardless of movement".

If I understoond you correctly, this is what you're asking for.

Can also be found as first result when googleling "LibGDX move particles with emitter".

Not familiar with LibGDX Particle Emitters especially, but according to this site one of the options for a particle is attached, which makes the

particles stay the same distance away from the emitter regardless of movement

This is somewhat confirmed by the official github-wiki, though the documentation is not 100% clear too me regarding this point. The Options are described at the bottom of the page just before loading Particle Effects in your game and described as:

Attached: Means existing particles will move when the emitter moves.

If I understoond you correctly, this is what you're asking for.

For your information: Can also be found as first result when googleling "LibGDX move particles with emitter".