At the risk of being a contrarian, Lucien, I have an Olympus E-M1.1 - and it has become my favorite camera for birds and birding. If there is sufficient light - and if I am close enough and have a quick enough shutter speed - the camera has continued to give me results which I consider quite decent.
Here are a pair of ducks, racing across Lake Euwana, outside of Klamath Falls, Oregon--
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Here are some Canada geese, one of whom I caught in mid-flap, in an irrigation pond near my old farmhouse in rural southern Oregon---
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Here is a soaring red-tailed hawk, in the plains and high desert east of Bend, Oregon---
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And here is a flying blackbird, in the reeds that surround a local pond---
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All of the preceding were taken with the relatively inexpensive 'consumer-grade' Olympus 75-300mm telephoto zoom lens. Obviously having a decent telephoto is an advantage, in that it allows the photographer to get a bit closer to their subject matter.
When I first got my E-M1 v.1, I spent some time looking at and studying the photographs taken by a number of relatively experienced wildlife and birding photographers, most of whom seem to be currently shooting with the much more capable newer OM-1 Olympus camera, and the significantly more expensive Olympus Zuiko 300mm f/4 prime lens. But the really interesting part, to me at least, was-- half a dozen and more years ago, many of them were also shooting with E-M1.1 - and were regularly creating phenomenal photographs with the same camera that you have.
Go figure