That'll be because the battery in your old controller is a bit battered, even though you wouldn't notice the gradual change. Compared to my original PS4 controller, the new one lasts twice as long - but I bought a new PS4 controller a couple of months ago, and that one lasts the same amount of time.
Right, there's a whole interesting subplot here, that pissed me off a bit until I realised there's nothing Sony can actually do about it, short of swapping HDMI for Displayport and rendering all 4K TVs useless. (I know you didn't ask about this specifically, but you reminded me that I wanted to pass on this warning!)
On the display settings menu, with a 4K TV, you should have two options:
(1) 2160p YUV420
(2) 2160p RGB
Whichever you choose, you're sacrificing something.
(1) is HDR-compatible, but gives you 4:2:0 chroma (i.e. compression, which gives you a slight loss of colour accuracy).
(2) is NOT HDR-compatible, but gives you 4:4:4 chroma (no compression).
For this reason, I would
set your display choice to "automatic". The theory is, when a game is HDR-compatible, it will switch to mode 1, and when it isn't, mode 2, so you get the best possible combination.
All of this is because, even with the included HDMI cable, HDMI doesn't have the bandwidth to pack in both HDR and 4:4:4 chroma together.
That's the basics - there's more about it over at NeoGAF, if you're technically minded enough to care (8-bit/10-bit colour, refresh rates and more):
NeoGAF: PS4 Pro, 2160p-YUV420 or 2160p-RGB
But, I'm glad I know.
(To answer your original question - if you leave it on "automatic" and then play a 1080p game, the Pro will upscale it to the maximum resolution of your screen, i.e. one of the above modes. This upscaling will look better than the one that your TV does with 1080p content, but not
massively different in most cases.)