(We split this thread from a question on the Italy board: http://chowhound.chow.com/topics/758532 You can find the question it was a reply to on that thread. The Chowhound Team )
I am not an expert on Italy, so I will confine my remarks to France.
It is not hard to eat all day in France. Like many places in Europe, France is very slowly converting to all-day openings for restaurants (look for signs that say "service continu"). You can get food at almost any hour (not the middle of the night, obviously) in Lyon, Nice and Marseille. The smaller the town, the less likely it is you'll find food at "off" hours (French people eat lunch from 12:00 to 14:00, apéro from about 17:30 to 19:00, and then dinner from 20:00-22:00, a great deal earlier than further south).
You can always get provisions, however, and bakeries remain open all day, though they bake bread first thing in the morning and around 17:00 for the dinner rush. The price of bread is almost fixed throughout France; it should cost 0,90€ (up to 1,15€ in fancier shops) for a standard baguette. France has a fairly lively street food scene; there are, for example, crêpiers plying the Parc de la Tête d'Or in Lyon, and any large park is likely to have a buvette (outdoor bar).
In fact, I would suggest Lyon for your French stop. Not that I don't love Marseille and Nice, but Lyon is the gastronomic capital of France for a reason. While it has a comprehensive (and cheap) métro, you can walk across the city—literally—in thirty or forty minutes of concerted effort. You can take the funiculaire to Fourvière, but the reason to go to Lyon is Presqu'Île, the little spit of land in the centre (2nd arrondissement) between the Saône and the Rhône. It's a place where people are very nice and friendly, and while there are plenty of things to see, it's not like Paris where you may feel cheated if you don't go to the various tourist sites. Lyon is a perfect place to wander and eat and drink.
There is an outstanding daily outdoor market in the place St.-Antoine, every morning until about noon. Beautiful cheeses (try a St.-Marcellin), fantastic fruits and vegetables (some local, some not, so ask before buying). Amazing bread, and of course no market in Lyon would be complete without sausage. At the upstream (north) end of the market, right on the river Saône, is a café that every Sunday morning has beautiful oysters. You can sit in the sun watching the river go by and dine on a half-dozen or dozen oysters with a Petit Chablis or a Sancerre.
In the evening, before dinnertime, you can stop in at any bar and have a kir (Aligoté and crème de cassis) or kir royale (Champagne and crème de cassis), or maybe stop on a hot day at a bar, sit down and have a snack and a glass of anis.
The nice thing about Lyon is that everyone, from the concierges at the hotels to the dustmen, is a born gastronome. You can stop into a bookstore and ask for a recommendation and the people will give it freely. Just avoid all the "bouchons lyonnais" in the Place St.-Jean; even one street up on the rue du Bœuf will net you better results for all your traditional offal needs. Quenelles de brochet, groins d'âne, tarte aux pralines, and of course heavy-bottomed "pot" after heavy-bottomed "pot" of local wines. Prices for three courses at a real bouchon are typically 19€-30€, depending on what you choose, and a "pot" of wine shouldn't be more than a few euros on top.
I may be in the minority, but Dijon is not the best of Burgundy. The historic centre is, sad to say, un piège à touristes—a tourist trap. Sure, the food is quite good, and the scenery is of course beautiful, but it doesn't begin to compare with Lyon in my book, and I wouldn't say it's possible to wander Dijon and just eat and eat and eat and eat the way you can in Lyon. (And yet, on my last trip to Lyon I lost 6 kg in three weeks—despite eating and drinking like a king.)