If all you are concerned with is a fast frame rate, slr may be the wrong place to be looking.
My Nikon D3200 DSLR has a max continiouse frame rate of 4 frames a second.
That's max frame rate. If the shutter speed is too low, it wont manage that, nor will it sustain it, buffer will allow perhaps 20seconds of continiouse shooting till buffering under-run sees it 'stall', at least at max image res.
My O/H's L310 Point & press bridge...... I dont know what the quoted frame rate is, but put it on C and tried it, and I got about a dozen frames in the time my camera took to get three! Not having the mirror mechanism to shift, having a lower resolution, having a lot less processing in the camera, or the processing more optimised for fixed programs, it works faster.
Her L310? £120. less than a 1/3 the price of my £400ish D3200. Takes photo's of near the same resolution, and out the box, has more zoom reach to its lens, than my kit 18-55, and unless you want to get very serious... which talking 'entry' SLR's is mutable......and get the know-how to exploit the extra manual control and versatility of a change-lens SLR... suggestion may be taking you down a perverse avenue, likely to NOT actually get you 'better' pictures, simply by giving you less 'out the box' capability for the sake of more manual control..... that might help an expert get a 'better' pic, but as like to give the novice more chance to cock one up!
Common problems going up to 'better' cameras in expectation they will give you 'better' pictures.
Long lenses. Fills the frame better with a distant subject. Your typical 3x zoom compact gives you a snap, with your horse a hundred yards away, filling barely 1/5 the middle of the frame, and a lot of unwanted field around it. Use digital zoom, or crop in on the original image in photo-editor.... that little bit of interest doesn't enlarge well, or show the detail you hope for in riders expression, and looses clarity.... but it IS there, and it is fairly sharp.
The small zoom, meant that you did get the subject in frame. may not have been perfectly centred, but it was in frame and you didn;t cut off the riders hat or horses tail or anything.
Big area of view, meant the camera got a lot of light to expose the frame, might have used a wide auto-apature, but at mild focal lengths probably focasing near infinity anyway, this didn't make focusing too critical, and allowed a high shutter speed that froze action.
Zoom in, movement in the frame is magnified. Any camera shake is going to cause blur, so you need to use a higher shutter speed; old rule of thumb was closest shutter speed to focal length.... so on a 35mm slr, with a wide angle 35mm lens, 1/30th of a second; standard 50mm, 1/60th second, 210mm telephoto.... 1/250th. t6hat's three 'stops' faster shutter on the telephoto, so you need three stops more appature, if you have them... and if you do, that will shorten your depth of field, or focus tolerence, so increase your liklihood of getting a blurry shot, which using auto-focus camera is likely to be less accurate given a distant and moving subject anyway..... or means upping your ISO setting, which will increase noise and reduce image crispness.
Ie: the 'better' camera, even if you know what its doing for you, is shortening your chances of getting the 'better' picture you hope for.... and if you only half know how to use it, and use it badly, shortening it further still.
So whats the purpose of the picture? You want 'nice' family album pictures, then going DSLR at the entry level, and facing the learning curve of getting to grips with how to use it and use it well, you are likely to loose an awful lot of pics that you actually want, in the learning curve, from the shortened odds of the more 'demanding' equipment.
So, I'd start with Moreorless's list of queries, about where you are starting. I have mentioned some technical terms in there, shutter speed, aparture, ISO setting, lens length. Depth of Focus, camera shake, blurr.... are you familiar with any of these? do you know what they are? What they do? Whats your 'level' of photography? What sort of pjotography are you doing already? Are you a point & press happy snapper (no shame in that! BTW) or are you an F-Stop anorak, to whom the bakan quality of apature blades is important?
Why do you want to take these pictures? Who for? How are they going to be viewed? And how 'good' technically do they have to be? Whats more important? The crispness of the image, or capturing 'the moment'?
What aspirations might you have to get more into photography? And how much is it really worth?
We dont have SO much to go on, but if as I suspect; you basically arent a keen photographer already, and you just want to get closer to your subject and get the crucial moment in a pic, that's probably NOT going to go into a magazine or anything, and at best be veiwed on a computer screen or digital photo-frame, by the family.... THEN a more non enthusiast, point & press compact or bridge camera, is possibly more likely to do what you actually hope, and give, more often, the pictures you want to capture, more reliably, and with less effort, and do it more cheaply, than an entry DSLR.
OH's bridge, I know would be good for probably 90% of the family shots I am every likely to take.
Very easy to be 'over-sold' a camera, and lead to believe you need more than you do, and that better camera = better pictures. It doesn't. Better pictures come from looking through the view finder and finding the interest. No camera does that for you.... any camera that lets you grab that interest when you see it, is a good camera...
A higher end compact, or bridge, therfore MAY be the more appropriate tool for the job you want to do, not an entry SLR.
