The bacteria that will digest your material need to have enough of a Nitrogen source to provide the protein their bodies need to digest the Carbon material they work on and the optimal ratio found for that is 1 part N to 30 parts C. so you do need to add a certain ratio of Greens (usually consider a high N source) to Browns (usually considered a high C source) of about 1 part Greens to 3 parts browns.

I think it will be fine to add material whenever you have it, keeping in mind to add some greens and browns. Eventually you'll want to let it "finish out" but that can be later when it gets heavy to turn.

Several years ago I had a compost tumbler and I threw everything into it. Each week when I cut the grass, any plant prunings, kitchen scraps, leaves as they fell, etc. I put stuff in as it was available (or as generated) and in several years it never got full. I just kept adding to it. The instructions say to go out and "tumble" (or rotate) it every day. I did that and each day added what I had into it. My family members jokingly refered to it as "The Black Hole"

I bought a tumbler and used it for a year and a half. Now I only use seven mesh bins.

For the quickest results, i.e. the most heat, fill the tumbler completely up with a good balance of nitrogen to carbon materials in one day. Then two or three days later, when it has shrunk somewhat, add a material that breaks down quickly, such as fresh cut grass or coffee grounds. Use enough to fill it back up.

Don't expect 14 day compost, I never got it. Some tumbler users say they can get it, my favorite book "Easy Compost" says to age compost 30 days even after it looks done.

I will say we're not looking for "quick" compost or anything...we'd just like to be able to constantly add to it instead of having to freeze stuff while we're waiting for the current compost to finish.

We're fine with adding stuff and then once we're ready, let it sit for a few weeks...we just want to know the best way to do it.

I'd like to suggest a mesh compost bin as well as your tumbler.

You can collect and store your fall leaves, or have another batch of compost in it. I would be much more confident to have problematic ingredients such as fish parts, manure, or veggie and fruit scraps in the center of 4' diameter bin 24" high than in a tumbler. In the pile you can keep the smelly ingredients in the core of the pile, whereas the tumbler mixes all the ingredients up.

I use a compost tumber also; actually, I use 2 tumblers. Last summer, I used only one and added materials as they became available until the composter got too heavy to turn easily (it was not completely full, just too heavy). That probably took til the end of summer. Then, in April of this year, I found I had great compost from the tumbler.

The problem with this was what to do with all of the backlog of compost materials after filling up the first? Using a 2nd tumbler has been a great solution for me, having a "spare" that can get all of the extra while one tumbler "is focused" on composting.

@tomindenver: When you continuously added materials, did you do it in any sort of ratio (greens vs browns)? Or did you just add whatever/whenever after the initial load?

I tried to add ingredients to keep the C:N ratio around 30:1. All my ingredients were weighed and put into my C:N spreadsheet.

As I mentioned in a previous post, things like fresh cut grass and used coffee grounds were used to refill the tumbler.

I tried to add about a one green to two brown ratio by volume, as that seems to work the best. I am also careful about adding water as the tumbler stays pretty moist on its own.