I'm making tonkotsu ramen broth, cooked over twelve hours (some people cook it even longer). I'd rather not leave my gas stove on overnight, so I'm wondering whether the same effect - breaking down collagen in meat and bones - can be achieved by cooking the broth in intervals of four hours over a few days. The broth would be refrigerated in between. It's not ideal, but I would think that the same collagen-gelatin conversion would take place... right? Is there a downside to trying to slow cook in short intervals? If this were braising meat, the meat would dry more, of course. But since we're mostly talking liquid, bones and fat, the change in temperature (and congealing then liquifying of fats and collagen) shouldn't make a big difference right?
The obvious downside is safety: each time you go through a heating and cooling cycle, your food will spend time in the danger zone. It's hard to guess exactly how long, but if it's a large volume and it takes an hour to cool to 40F and 15 minutes to heat back up to 140F, and you use the conservative end of the safe time range (2 hours) you might have a problem. You could partially alleviate this by using an ice bath to chill faster.
If you can avoid the safety issue, in terms of the resulting broth, it sounds fine though.
That said, slow cookers are awesome, they're not too expensive, and that'd solve your problem with much less effort.
Jefromi's stated the important food safety issue.
I do think there could also be differences in the way the food cooks and breaks down. One thing I'm thinking of here is the way gelatin dissolves and breaks down differently when reheated compared to when it cooks originally. Stews taste differently on the second day partly because of the way the gelatin, well, gels at low temperature and then may not redissolve completely upon (relatively quick) reheating.
Obviously if you cook the food long enough over a few days, it will break everything down. But I would wonder whether there might be any alterations in texture or flavor due to the temperature cycling.
I also imagine these cycles of heating and cooling will also affect how various aromatics, spices, herbs, or other ingredients affect the final dish. The extra time and processes that go into heating and cooling (especially the thickening and melting/redissolving) may affect the relative strength of various flavors, some being lost or "overcooked" and others enhanced. This probably wouldn't be a major issue, but it could make a small difference.
Is slow cooking in an oven safe with long cooking times might be relevant or even, for this task, a viable method. An oven offers some thermal buffering even if the heating element cycles, and can be monitored with an oven thermometer. Bringing the contents to the same temperature the oven is set to before putting them in might help.