Missing space on device storage

[Steve Swift]

My wife's S4 Mini has run out of space in the device storage. We can't even install a tiny app such as Disk Usage (hoping to find where the space has gone).
So we moved a couple of large files (about 600Mb in total) from device storage to the SD card (at least 32Gb, probably 64Gb). This made absolutely no difference to the free space on the device storage.
So, can anyone tell us what's going on here? Is something filling up the device storage as fast as we try to clear it?

 

[Steve Swift]

Additional information:

Using the Storage details (in settings, more) we can see that "Miscellaneous files" are consuming more than 3Gb. Yet when we sum the sizes of the files, there's less than 1Gb accounted for. We're seeing hidden files (well, ones that start with a period; are there TRULY hidden files in Android?

Incidentally, is there a way to install an app from Google Play directly onto the SD card?