I have the Sony FS700R, EA50U and A6000.
I want a longer lens for some situations and am looking at Sony 70-400 G2 or Sigma (Bigma) 50-500.
I am leaning towards the Sony A mount with LA-EA3 or LA-EA4 adapter.
I want it for photos and video.
I would like to have the option to use autofocus when I need it. Stabilization would be great, too.

It's my understanding that on capable cameras, the LA-EA3 is preffered. But some Sony cameras cannot focus with some lenses and thus need the LA-EA4 which has it's own focus system.
I do not know if the cameras I have need the EA4 or I can get by with the cheaper EA3.

Would the Sigma with Metabones be quite inferior to the Sony as far as focussing?

Thank you for any help. I'm still trying to learn about interchangeable lenses

David I have a FS100, EA50, and A6000 and use either EF mount or Amount with LAEA3 or LAEA4 adapter. I also ahve a few EF lenses with Metabones adapter as well.

If I need auto focus with non EF lens then I use A mount with LAEA4 adapter. It is very responsive, however due to the AF mirror inside the LAEA4 all lenses will be stopped down when using AF. If you use push AF then your lens will switch to 3.5 and then back to selected faster f/stop. If you have continuous AF on then you will only get as fast as 3.5 on your f/stop.

However when I need more light and am fine with manual focus then I use the LAEA3 adapter which let's more light in anyway than the LAEA4 and will stay at a fast f/stop like 1.4 or 1.8.

However if you want continuous light and AF then you could go Metabones route with lets say EF mount. However the Metabones is not as fast or accurate as the Sony mount. Personally speaking I like the Sony adapters better and recommend both if you can for different situations. I use these with Sigma glass.

Although the downside to A-mount glass is that you can sell EF mount glass easier than A-mount due to popularity. As Canon or Nikon mounts have more demand and resale value.

Thank you Michael. That was a detailed and helpful post.

This video might help. https://youtu.be/hvdQkiMqz5s?t=2

Thank you Mervyn.
That video did help.

I wonder how the Sony e-mount 70-200 2.8 (with teleconverter) would compare with the Sony a-mount 70-400 (with adapter) in terms of loss of light and autofocus.

The 70-200 2.8 with converter could have as much reach, yet lose the wide-ness and possibly be faster focussing and still more light sensitive??