We were near there two years ago during the heatwave that killed all the old people (40+ all day and never below 30 at night) - there are loads of cave dwellings all around the area which are brilliantly cool when it's hot and I guess would hold some interest for your DCs - show them the Flintstones before you go.
My Irish blood is usually the house wine for French insects, but a liberal spray of Neal's Yard citronella stuff on exposed skin every evening kept them pretty much away for the whole fortnight - recommended.
We were in that area about 5yrs ago. It's gorgeous. As BBBH says, there's lots of caves and things to visit - try Les Eyzies and locale. Domme is good (road train to ride on) as is Sarlat and also Perigeaux - market towns to wander around. We didn't have any problems at all with midges.
Look out for signs saying 'XXXX (village name) en fete'. These are usually wonderful occasions when the whole village turns out for an open-air meal, knees-up etc. Multi-generational family groups are the norm at these events and we have found them to be extremely friendly. Check out the local 'Syndicat d'Initiative' (tourist office) for details. Most regions also publish a seasonal newsletter giving details of cultural events, child-friendly places to visit etc. The Syndicat d'Initiative will have one.
If you're staying in Sarlat, then you must take a trip to Beynac which is about 15 minutes away. It's a medieval village on the river with a chateau at the top of a very steep hill. You can either park up at the car park at the bottom and walk up, or drive up. We stayed, last year in a cottage near the chateau gates and it was wonderful. YOu must go the restaraunt in Beynac on the right hand side of the cobbled steet leading up to the chateau called Le Petit Tonelle where they do the most fabulous food which is French but with an oriental twist as the husband and wife team are French and oriental.
They are great with children. We have been going there for the last 3 years starting with when dd was still a little bump in my tum.
Activities wise, there are loads of caves to visit, most of which are fine to take children into. Try and get to Le Gouffre de Padirac on the Dordogne/Lot border which is to the South East of Sarlat about an hour away. This is a fabulous trip into the caves, where you are taken round part of it on a boat. Your children will love it.
You must also go to Le Bugue where there is a brilliant aquarium here and right next door, Le Village Du Bournet which is an open air musuem recreating life in the Perigord region in the 1900's. They have loads of stugg for little children to do with an old fashioned amusement park and demonsatrations of old crafts like tanning, glass blowing and wood turning. ds got a hand made spinning top for 1 euro last year.
Also Les Jardins de Marqueyssac where you can walk along the cliff top overlooking the Dordogne, visit the Marqueyssac chatuea and there are children's play park and horse drawn cariiage rides.
Sarlat town itself is very busy but quite alive at night in the peak season with street entertainers and you wont have a problem finding anywhere to eat in the pavement cafes.
Very touristy though and busy in peak season.
Le Roque Gageac is beautiful too and is a village set into the rocks above the lovely Dordogne river.
It goes without saying that you should take a trip on the river, either on one of the tourist boats or on a canoe - just be careful, you don't canoe the wrong way down the river as my dh did when I was 8 months pregnant. He grounded the boat and I ended up trying to pull the boat, him and ds out of the river with my huge bump sticking out.
A bit further afield and worth a visit are St Cirq Le Popie, Rocamadour (although extremely tourist and a bit tacky now) and Peche Mel caves where you can descend into the caves in a balloon basket.
Domme is nice but very busy. Best to go early evening when it quietens down a bit - th shops will still be open and there are some good open air restaurants. Avoid Le Poivre restauraunt. Looks nice but food is shocking. The restaurant right at the top of the village overlooking the Dordogne valley (next to the carousel) is good.
Oh blimey, sorry, I never saw this! Haven't got much to offer as I went there BK (Before Kids) but the river is a must and I loved Sarlat market - not sure if it's every day though. There's a produce market and also a clothes market across the road. There's a fantastic children's clothes shop on the market square where I got a lovely raspberry red sweater with grey trimmings for my friend's daughter.
We cycled to Rocamadour from somewhere about 30km away which was incredibly hard work but amazing scenery, but Rocamadour itself was a bit of a let down really.
I have immensely happy memories of the Dordogne - sorry I can't help more re kids stuff.