Nowadays, a miner can gain more than another one even when they mine the same amount of blocks due to variance in tx fees. It's expected that, especially once the coinbase reward is gone, this will result in miners competing for "juicy" txs and hence damaging the stability of the system. See https://dl.acm.org/citation.cfm?id=2978408 for future reference.
I've been thinking if it's possible to overcome this simply by changing the rewarding mechanisms of Bitcoin. So let's say we keep a k-length jumping window, take sum of fees in that window and divide it evenly to miners. So if we have k=4 and 1st block has 20 in fees, 2nd has 20 in fees, 3rd has 10 in fees and 4th has 30 in fees each miner will get (20+20+10+30)/4 = 20 in fees.
What would be shortcomings/downsides of such a mechanism ?