Hi, I've been lurking for a bit and this is my first post here. I knew exactly what you meant because I do the something similar. I was told by one of the old locals that he actually finds it harder to hunt by camera than rifle...
My weapon of choice is a Canon 7D MKII with a Sigma 150-600mm lens. I like to shoot birds and they can be quite small subjects. I shoot handheld and it took quite a bit of work to be able to track a flying bird at 600mm but it is possible. I shoot in burst when they move fast usually at shutter priority of 1/800 and ISO set to auto with a max of 1600.