One thing you will lose is the sensor stabilization. The A-mount lenses series of lenses rely on the camera body to reduce the effects of hand tremors when shooting hand held. I've used my 70-400 Sony Zoom on the NEX-6, 7, and the Alpha 7R. Autofocus on these cameras is faster than what I can achieve with their E-mount lenses. However, handholding over 200 mm requires watching ones shutter speed and you may wish to use a tripod with the longer focal lengths. For the Normal and wide angle lenses, I do not find the lack of stabilization to be too much of a handicap.
Also, since you are phase detectors in the lens adapter, there is a chance that a lens may back or front focus. When I use the LA-EA4 adapter, the menu for correcting for this becomes active. Unfortunately, I do not know for certain if this occurs for the A-6000 but if it does, it is a worthwhile command to keep in mind. Having said that, I have used three A mount lenses on the NEX mount, a Tamron 90mm, a Sony 16-50mm, a Sony 70-400mm, and a Minolta 500mm f/8. All of these lenses did not require correction. So I've been lucky in this regard.