I haven't used stainless so I can't address it.

I have a few cast iron items, one by Le Creuset and two by its imitators--an enameled Lodge ('Lodge Color')and one by 'Best Equipped Kitchen'. I've also used a Lodge raw cast iron (non-enameled).

The Le Creuset cooks very well and is gorgeous. It is pricey, though, and you have to ask if it's worth the premium over the imitators. I have to say, I've been happy with the performance of the cheapos. Now, will the enamel last as long as LC? Dunno, but since they are about a fourth or a third of the price of the LC, the LC would have to last 3 or 4 times as long as the cheapies to justify the premium, IMHO.

Of course, if you don't mind paying that premium by all means go ahead--LC is a great item.

If you want to try out one for size and applicability, you might want to go by your local Sam's Club; I was at my local one yesterday, and they had a 6 qt round Dutch oven for about $40. And although it had the 'Members Mark' label on it, it sure looked suspiciously like the Lodge Color Enamel. I wouldn't be surprised if they rolled off the same Chinese assembly line. Another source for inexpensive items is TJMaxx; I got my one piece of LC there because it was a second (a slight swirl in the enamel of the lid, doesn't affect the performance a bit).

I have both oval and round; oval is nice when you want to cook a whole bird in it, round is nice as a general stovetop pot--both cook well. Sizewise, bear in mind that these suckers are HEAVY. A 9 qt filled with food will be pretty hefty.

I'd probably go with two pots, one a 2.5 to 4 qt and another in the 6 to 8 qt range.

And if you want to put these through their paces, by all means get a copy of Molly Steven's excellent book 'All About Braising'. Wonderful info.

Here is a link that might be useful: all about braising

"What are the pros and cons of stainless (e.g. Allclad) vs cast iron (e.g. LeCreuset)?"

A true Dutch oven is made from either bare cast iron or enameled cast iron. From Wikipedia:
"A Dutch oven is a thick-walled iron (usually cast iron) cooking pot with a tight-fitting lid."
"Modern Dutch ovens designed for use on the cooktop or in the oven are typically smooth-bottomed. Two French manufacturers of enameled Dutch ovens, Le Creuset and Le Chasseur, refer to their ovens as "French ovens", or in the UK as "casserole dishes". Some older styles, such as the unglazed ovens by Lodge, CampChef, and Wagner, retain the bale handle, while others, such as the enameled versions by Staub, Sante, and le Creuset, have two loop handles."

Stainless steel is not really efficient for true slow-cooking. I would look at TJ Maxx or Marshall's for less expensive Le Creuset, Nomar or Staub. I'd be leery of made in China pieces regarding quality. I have an older Martha Stewart one made in China and it is not wearing well. I'd put my $$ on Le Creuset, Staub or Nomar. they're definitely worth the price and you'll have them until your last breath ;o)

We cook almost exclusively with pure convection. Would this influence the choice between stainless steel and cast iron?

The convection cooking would not really affect cooking in cast iron. One of the main reasons for buying a Dutch oven is for long slow braising or stewing of meats, root vegetables, and foods needing long cooking like dried beans. Convection cooking promotes more even oven temperatures in all areas of the oven, which can provide faster and more even baking of things like cookies. But when cooking long, slow, and low in a cast iron dutch oven you might just be wasting energy to have the convection fan run the entire time.

This is just my thinking - someone else may have a different take.

Another point, a stainless steel pan lid may not keep in moisture during long cooking as well as a cast iron lid. This is why I cook my grains (rice, barley, etc.) in my LC cast iron French oven rather than in my Calphalon saucepan.
Teresa