Difference in aperture and sensor size - anyone willing to help

Ok so I need someone to do be a favour here since I don't have the gear to test myself.

What I'd be interested in visually seeing is the difference in background blurring between a fullframe camera with a 135mm f2 lens and a crop sensor shot at f2.8 over the same focal length whilst taken in the same physical position of the same subject.

One thing that is plaguing me in some of the equine photography I'm doing is that the background, its full of distracting elements that are very hard to avoid in the setting (esp indoors). So I'm wondering if a shift in equipment might give at least a more pleasing result over what one might consider a typical arena.
I'd rather not sit for hours making masks and using selective blurring to enhance what my shot already has and would rather get it in camera (because that takes seconds - whilst the editing could take ages).

If anyone has done this (or similar) I'd be eager to hear of your real world experiences and if you found any difference worth noting.

If you use a lens for a full frame sensor on a crop sensor, I would not expect to see any difference in the images.

135 mm on a FF and the same focal length on a crop, from the same position? Apples and oranges - your field of view is going to be different on the two of them. If you crop the FF down to match the framing on the crop, the difference in DoF will be around 1/2 the width of a horse at 50 ft. (and f/2 would give you a DoF of just over the horse's width). The bokeh will depend on the lens(es) used however, and that may be something to consider as well. But you'd want either an 210 on FF to match your 135 on a crop, or 85 on crop to match the 135 on FF for field of view.

Edited to fix flipped focal lengths.

FF and crop are different formats. So as Jon already mentioned, you will need to change the focal length of the lens to get a fair comparison.

Smaller cameras need smaller lenses.

If you shoot your full frame at f/2.8 with a 135mm lens at (for instance) ISO 400, then the crop camera should be at f/2 with an 85mm lens at ISO 200. The shutter speed stays the same. Then the field of view, DOF, motion blur and noise should all be approximately equal.

There will still be small differences depending on the idiosyncrasies of the lenses used.