Handbrake is an excellent converter.
It appears to be mis-interpreting the footage - these are definitely interlacing artifacts. Are you sure you have your camera settings right?
***EDIT***
I just looked up the A7 specs. It has a setting for 1080/60i. You don't want that. It also has a setting for 1080/60p. That's the one you want. I have no idea why Sony would include an i setting. That's crazy, IMHO.
**********
The first thing to try is any updates to Handbrake. It may be that the latest version will clear up this problem.
Otherwise, you'll need to go into the Handbrake picture settings, and try some of the deinterlace controls. Here is the official guide:
https://trac.handbrake.fr/wiki/DeinterlacingGuide
The guide explains the controls, and also provides links to further resources.
Here is more on decombing vs. deinterlacing:
https://trac.handbrake.fr/wiki/Decomb
And here is a general orientation to the Picture Settings controls:
https://trac.handbrake.fr/wiki/PictureSettings
Finally, there may be some A7 settings to explore. As you've found, only the moving elements of a scene will show interlacing artifacts, bear this in mind when shooting tests. I'm not familiar with the A7, but here are some general issues which may crop up:
* Some cameras will embed 24p in a 60i wrapper. The resulting video IS 60i, but looks like 24p. If an encoder like Handbrake is given 60i, it will convert to progressive. Test 720p30, 1080p60, and 1080p30 and see if it clears up. If so, the immediate solution is to shoot in one of these modes. If you simply must shoot 24p and your cam will only do it in a 60i wrapper, you'll want to investigate the de-telecine settings in Handbrake:
https://trac.handbrake.fr/wiki/Telecine
.
* Likewise, some cameras will embed 30p in a 60i wrapper. Perhaps the setting will be called something like 30psf. Here you may not have many choices of camera settings; you'll want to investigate the de-telecine settings in Handbrake:
https://trac.handbrake.fr/wiki/Telecine
.
* Or, just skip shooting video with the A7. It's old 
Please post back any solutions you find...
From the upper left corner of the U.S.
Photos, Video & Pano r us.
College and workshop instructor in video and audio.
70D, Sigma 8mm, Tokina f2.8 11-16, Canon EF-S f2.8 17-55, Sigma f2.8 50-150 EX OS, Tamron 150-600VC. Gigapan Epic Pro, Nodal Ninja 5 & R10.