As far as *I* can tell, the issues herein (plural) are considerably more complex than many people seem to realize, and pretty much ALL streaming devices that are capable of “auto-adjusting” the frame rate suffer similar if not identical issues! So regardless of whether you have a Roku, Fire device, etc. it all boils down to this:
For proper handling of frame rate changes and related a/v syncing, it all depends on your tv’s fps-changing capabilities and settings (some tv’s can handle the changes both accurately and very quickly while others are so slow and otherwise limited at it that hdmi error messages are generated), fps changes from one content source to another during a streaming session (e.g. switching between paid advertising “commercials,” device UI, and primary program), and even your tv’s hdmi version and the capabilities built into the streaming devices’ apps or “channels”!
There is in fact very little practical difference between device hardware brands in this regard, and I don’t see the situation changing much until greater standardization across hardware components and brand models, software, and content sources comes into existence. Chances of this happening soon? In my estimation - SLIM.
A FEW streaming SERVICES - not many, but a FEW - have thus managed to create a work-around for themselves that involves detecting a potential program content change-inducing fps change IN ADVANCE, and ALMOST IMMEDIATELY followed by an automatically created fps change, such that the problems created by the changes back and forth are at least somewhat less annoying than they otherwise would be, while some OTHER services have simply settled on seeing to it that the frame rate delivered to your device (and thus tv) remains CONSTANT (with any changes in fps needed to provide consistently high video quality and proper a/v syncing being performed BY THE SERVICE ITSELF IN ITS “BACK END”). - Notice how Disney+ always seems to look and sound so perfect regardless of whether or not you’re viewing a television show or theatrical film, for example?
So it all depends on one’s particular combination of things - MANY “moving parts”! And so for the time being, at least, most but not all will probably find that if one’s streaming device’s fps (or rather the device’s signaling of fps to the tv) can be manually set to either auto-adjust on or off that OFF will usually produce the least annoying outcome (in USA that’s normally a consistent 60 fps, which US television service in general has been built around).
