If you want to do that you would want to use a crossover, not a bunch of EQs - not that I would recommend it in the first place, unless you're talking live sound, then yeah, been there done that (with crossovers, that is).

Your concept will work if you make each equalizer's frequency selection to overlap each others. You WOULD NOT want to daisy chain them in series. But rather, you need to sum them into separate mixer inputs so that the output is a summation of the numerous inputs. Series daisy chaining will probably provide you with the worst sound you have ever heard.

Yes, an electronic speaker crossover, whose outputs individually feed individual compressors and then summed into a mixer as described above, is really the way to go.

But what do you really hope to accomplish with this? Automatic mixing? Great in concept. Poor in execution. Great for broadcast loudness maximization as Mr. Durough made so popular back in the late 1970s with his 3 band discriminate audio processor did. And that Orban and CRL expanded upon and also made popular for broadcast purposes. Otherwise, I think you're barking up the wrong tree?

Your dog just peaed on my foot!
Ms. Remy Ann David

The only application where I've seen decent use of multiband compression has been in broadcasting, and companies like Aphex and Orban manufacture gear to address that niche.
Bent is right in that the use of an active crossover ahead of the compressor is how to do it, not by merely using a sidechain EQ. Personally I use a sidechain EQ all the time for frequency-dependent compression, but that's not multiband.

I want to buy 3 or 4 of the same type compressor and 3 or 4 of the same type EQ.

Additionally, that is a crapload of gear ($$$).
Most of the EQs I play with (Yamaha, Ashly, Klark, etc.) only go (via switch) +/- 6 or +/- 12dB - far from true band limiting, which makes it fairly pointless, the way I see it...

One box (as Moonbaby stated) can do it for a lot less.
Not to mention the fact that if you are mixing in the box there are plenty of multiband comp plugins that will get you what you want.

To reiterate my point: Outside of large sound systems (and broadcast facilities - thank you Remy), I don't particularly see the need for multiband comps; They make me feel like I'm chasing my tail more than anything else... (mastering folks might use them, but I don't delve that deep into their gig).

Sound Forage and others feature multiband compressor plug-ins. I think all of them sound like crap! Sometimes moderately useful as a "De-Esser" and/or for a "plosive" reducer.

Although I have rather liked the IK multimedia multiband limiter featured in "T-Racks" but I know a lot of folks here hate that program. But it does do the job however better than most of those other plug-ins. The VST version gives you more control over the multiband limiter than their standalone version which I use mostly. But that's just for some peak limiting and not compression for which it is over-the-top for.

I like T-Racks simply because I like Barney the purple dinosaur.
Ms. Remy Ann David

T-racks is a good piece of software.
One of the guys at work turned me on to it years ago.

Vegas is where it's at! You might remember where I stand on that - we've discussed it prior to this post, a few weeks ago...

I rarely use Sony's plugins, occasionally the default gate in vegas, but never the verbs, comps, etc.

removed

I guess this sounds like a naive question, but is the reception of your broadcast back to the station so bad that you can't record it off the air?
Maybe I was spoiled at the station I used to work at, but we could monitor and record a pretty good signal off the air. Then again, I could open a door and throw an RE-20 at a tower...
In any case, you might look at a used Aphex Dominator, which is a multiband processor. Very simple to use, very effective, and designed to be used in your situation. There is currently one on e-Bay (a 722, the 'fancier' one) for $299. I think that may be your answer.