Thanks for the videos. I sent you a PM to see if you could send over the dashboards IDs and allow us to take a closer look to see if there are any other suggestions we can make. 
Edit: also copying part of the PM into the main thread in case it helps anyone else who runs into a similar issue and finds this thread.
I noticed that a smaller tile size was being used, so I’d like to take a look at the overall number of tiles on the main (Living Room?) dashboard. Specifically, if there are a large number of small spacers used, those could contribute to the switching time on much older devices like this that have limited processing / memory. If that’s the case, you might be able to make some tweaks like consolidating several small spacer tiles into one large spacer tile as this reduces some of the calculations that have to be done between transitions.
As James noted, you might also try temporarily simplifying your backgrounds to simple colors (no images or advanced effects) and reducing the number of Media tiles for troubleshooting purposes.
Edit2: Just an update that Robert was able to get that dashboard transitions down to ~2 seconds on his old iPad Mini by removing a large number of unused Spacer tiles.
If you are prompted to remove excess spacer tiles when saving your dashboard, in general you’ll want to choose the option to delete the excess tiles. Similarly, if you have a large number of small spacer tiles, it’s often better to use one large spacer tile where you can.
While it likely would not have any noticeable performance impact on a modern device, a large number of tiles can increase the time it takes to transition between dashboards on less powerful devices. (In this case, it was something like 80% of the tiles were spacers, so this made a big difference on this 7+ year old device)