That really entirely depends on the brand / model. Some will overheat in under an hour, some will go on forever as long as you supply power. Price has nothing to do with it.
DSLRs will almost certainly be a cheaper option than a mirrorless camera for this, because pretty much any DSLR from the last ten or so years works for this, and there are many more of them. Autofocus and availability of clean HDMI in video-liveview mode depends though. Many support direct streaming over USB using proprietary tools.
> DSLRs will almost certainly be a cheaper option than a mirrorless camera for this, because pretty much any DSLR from the last ten or so years works for this, and there are many more of them
That used to be true, but with used a6300 cameras going for around 300€ nowadays, it's not the case anymore.
DSLRs will almost certainly be a cheaper option than a mirrorless camera for this, because pretty much any DSLR from the last ten or so years works for this, and there are many more of them. Autofocus and availability of clean HDMI in video-liveview mode depends though. Many support direct streaming over USB using proprietary tools.