As there are a few parents with young children I'd like to start this topic to help each other out with ideas for travelling with kids and visiting WHS. From my own experience I can come up with following ideas:

Visit WHS in Europe
If you still have some left in Europe you can do that with small children. Restaurants and Hotels are usually nice, there are good hospitals in case of an emergency and most countries are clean enough. You can do daytrips from the same place without the need to change hotel. There are so many WHS that drives are barely longer than 1 hour.

Go on a cruise
No need to change the hotel or even drive a lot. You have all you need on board the ship. Mediterranean or Caribbean cruises bring you to the next WHS while you sleep. For older kids there is even day care on board.

Island resort holidays
What most of us avoid at all costs is actually the most relaxing way to go on holidays with a small kid. But if you choose the right island, you can combine this with ticking off a few sites. We did Mauritius (2 sites and a twhs), Tenerife (2 sites and an easy day trip for a 3rd) as well as Ibiza, Mallorca and Rhodes (1 site each). There are more suitable islands like Cyprus, Gran Canaria, Greek islands, or possibly Crete in future.

Stick to expensive destinations
Usually the expensive countries are also those with better infrastructure and less health risks. Good places to go without risking the health of your kids and visiting lots of new WHS are e.g. Israel or Japan. Australia and the US are also suitable, but have a lower density of WHS.

Hey,
we just did a 12 days roundtrip on Sicily this March with our 4 months old daughter.
We originally planned to stay in max 3 different places, but ended up changing accomadation daily. This way, driving distances are shorter, because you only have to go in one direction and not back.
We started in Catania (2 nights), made a nice trainride around the Aetna, went then to Milazzo (2 night) with a dayrip to Vulcano, Lipari would be also suitable, then its just a short motorway drive to Cefalu and Palermo, from there we hid south to Agrigento, back into the mountains to Piazza Armerina, and spent then a few days in the Val di Noto. We chose Ragusa and lemonfarms near Noto to sleep.
Before we ended the tour we still visited Syracus. So in total, we visited 7 WHS sites in 12 days, The car rides comparably short and whileour baby was sleeping, we booked the nights spontaneously (hotels, Agriturismo, and AIrbnb), accomodation was cheap, rented car even cheaper, and in Sicilly the people love small Bambini, so you are welcome everywhere.
Our tip for traveling WHS with a little baby!!

I agree with kintante, and can highly recommend this as an easy option to see many sites affordably, with the ease of being able to stay in the same hotel room for the duration of the trip. There are generally pools and a variety of food options for kids, in addition to kid's play areas, on many cruise lines. In addition to the Mediterranean and Caribbean, there are also cruises to the Baltic, Australia and New Zealand, Alaska, the Panama Canal, and Atlantic Canada, amongst other options.

In order to maximize World Heritage Sites in the US or Canada, I might recommend hotspots like the Canadian Rockies near Calgary (five World Heritage Sites nearby), Atlantic Canada (three World Heritage Sites near Halifax, and another two or three farther afield), or the US Southwest (three World Heritage Sites near Albuquerque, and two more a little further afield). Additionally, New York City currently has two World Heritage Sites, and could gain up to three more depending on what happens with the US tentative list.

Another country for slightly older kids may be Jordan, which allows families the opportunity to camp in the desert at Wadi Rum, explore desert castles all around the country (including Quseir Amra), visit the spectacular ruins of Jerash (on the tentative list), and play Indiana Jones at Petra.

Thanks for the post above Jakob, I was looking around for essentially that information as I was looking into a potential trip to Sicily with two young children.

That was really helpful, out of interest were there any particular Agritourismos you would recommend?

Can I say that I loved @meltwaterfalls' review of Le Havre and showing the possibilities of finding fun for small children. I've had to become more inventive and find sites with boat trips, train rides and play areas attached over the past few years. But it shows that if there is enough out of the ordinary, kids' imaginations will take over. I never imagined it would be Le Havre that demonstrated this though (even before children I couldn't think of a persuasive enough argument to convince my girlfriend to visit while we were in Normandy!). Right kids, this way to the Struve Geodetic Arc!

Thanks Liam, it took a bit of preparation but worked out well, and the swimming pool at Les Baines des Docks was a really big help.

It also helps that it only actually a short drive for us to get there so the kids didn't get too bored, (in fact in terms of miles driven it is actually our closest WHS (30miles + 1mile from the ferry to core zone, it is even shorter if you are on foot)if you ignore the 6 hour ferry trip in the middle) also my wife had a genuine interest in the architecture too.

We have, only semi seriously, looked at driving the whole length of the Struve Geodetic Arc in a camper van, the Russian invasion of Ukraine put a bit of a spanner in the works, but I am not sure the kids would survive that one.