@trumee
100% everything @slampman said... 
if this is the one you have... LinkPi Enc1-v3P
All the LinkPis max out at 4K30 (if they are the ones that support 4K) so make sure you are setting your output from the firetv to that. If I tried to set it to something higher my Apple TV would change the output BACK to the 1080 setting since 4K60 is not supported.
I am running the 5 port version. I don't have the Internet speed to really test it...I drop frames even at 1 input set to 4k youtube stream... AND the encoder specs says max 3 - 4K30 encodes... I am not surprised if it bogs down at 2 though as @slampman says.
Here is a pic showing the HDMI-3 on mine set to 4k30 but I never run more than 1080-60 normally. Just as a side note don't forget these only encode output to stereo also.
Late Edit...
AS I look at the Note on the Amazon listing in this pic...
I see that they say your unit supports 1080P30x4 encoding/decoding..which they say is a reference for what can be encoded/decoded at once.
Which means your unit should support up to 2 - 1080P60 streams (1080P30x2 for each one) OR 1 - 4KP30 stream.
This would also make sense what @slampman said about our units struggling with more than 2 4K30 streams since our spec says 1080P30x8 EVEN THOUGH it also says it can encode 4KP30x3 right above/next to that on their website. 