Which model Amazon Fire Stick do you have ?
Does it max out at 1080p or 4K ?
Do you have the NOW Boost Membership and if so do you use 5.1 surround sound on it ?
From @Popeye13 workaround (which i appreciate he found it) and since using the workaround i have found setting my Fire Stick to its forced max resolution in my case 4K and when watching NOW use 50hz rather than 60hz.
Plus leaving my Boost surround settings under My Account on the Fire Stick NOW App to Stereo.
Not ideal if you want 5.1 when watching NOW.
To be honest i have never really watched Sky News on my Amazon Fire Stick or any NOW device (more a BBC News man myself) but if it has a news ticker bar moving across the screen then this is a stress test for smooth motion.
I found using some TV motion processing on my new television helps masking / reducing the amount of stuttering i see (but this only helps slightly on sports) where on other NOW content such has movies you obviously see the Soap Opera Effect which i hate.
What happens if you force 1080p 50hz (the Fire Stick should have this option instead of Auto or 1080p 60hz in it's Display Settings section).
Then when watching NOW on the Fire Stick and under the My Account NOW App on the Fire Stick when clicking on the Boost Membership setting button toggle between Stereo, to Surround Sound and then toggle back to Stereo once again and then leave it set to Stereo.
Does the micro stutter still remain when watching live Sky Sports ?
See if this above test works instead of dropping the picture resolution down to 720p when watching NOW.
Unless 720p 50hz is the only setting that helps for you by either reducing or eliminating the micro stutter on the NOW App with your third generation Fire Stick with your television set after toggling between Stereo, Surround Surround and back to Stereo.