Something that often gets missed in this is how autofocus will perform depending on reach. For a perched bird that is not terribly distant your results are not too different in image quality after crop nor in autofocus performance.
Having a crop sensor extends your lens's reach, and effectively makes your autofocus array larger. If you are relying on autofocus, a crop sensor will outperform a FF in many cases assuming you have similar autofocus arrays.
Translating that to real world - a 7DII will outperform your 60D in basically every way possible. A 5DIV will as well. However there are situations where the 7DII will outperform the 5DIV due to the intersection of lens reach and autofocus grid size. Here's a link to an article with an instructive image showing the difference in coverage of AF area:
http://www.the-digital-picture.com/comparisons/canon-eos-7d-ii-1d-x-5d-iii-af-comparison.aspx
The one strong advantage of FF cameras is for ultimate image quality and greater low light performance if you have a good enough / long enough lens and/or are close enough to your subject. For me, though, the weight and cost of a longer lens are just not worth it, I greatly prefer crop sensor and the extra reach without more weight.
In my opinion m4/3 cameras aren't there yet for autofocus performance for birds in flight. They are getting there, and I'm eagerly awaiting a year or two more development and hoping to replace my 7DII + 100-400 lens with a prosumer bridge camera. But not yet. For bird photography, I don't have any interest in a FF camera, and I wish I had a much lighter setup (ergo my desire for a bridge camera with better autofocus!).
From where you are at, kit wise, I would say the obvious upgrades are either a 100-400 mark II, or a 7DII. The lens will get your more reach and better IS. The 7DII will get your better low light performance, better autofocus, and better framerate. The 400/5.6 prime is a very sharp lens but having finally upgraded from that to the 100-400 mark II, I find the advantage of image stabilization so great that I would not recommend the 400/5.6 any longer unless you use a tripod a lot. Even in good light with very high shutter speeds, I get far more tack sharp photos from the 100-400 mark II.
Cheers,
Josh