Most campsites accept dogs (some wont) but be careful the more greedy ones will try and charge as much as £5 per night per dog (but do your homework and many are free)
I have caravanned with my dogs and Bruno my older one has been to virtually every European country we as Brits are allowed in and Amy has been through 3 countries now.
You need a pet passport to leave the UK with your pets and will take 6 to 8 month to get all the tests and vaccines done before they can travel.
They also need to be inspected by a vet on the Euro side 24 to 48 hour before your return to the UK for check over.
Driving you have to stop ever 4 hour or so to let your dog stretch his legs and other doggy things.
We gave up caravaning when we got the 2nd Great Dane as they literally were destroying the inside of the van moving around (furniture more fragile in caravan and euro stlyle motorhome) but again depend on your dogs and how good they are, mine are large 15 stone each.

Both travel superbly especially now they can sit up front with us and our Amy thinks she's in charge when arriving at sites (see picture

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Remember on sites to keep your dog on a lead and try not to let them have accidents in public areas and always clean up after them and you'll be fine.
I find the biggest problem with the dogs is attending the shows as mine are so large and at shows vehicles tend to park nearer to each other some mine can be a handful with other dogs around.
Saying all of this if the dogs cant go we dont either.