Why is Android 5 corrupting my SD card (moving files to LOST.DIR or losing them)?
You should probably mount the SD card with a USB reader in the PC, to see if the SD card or filesystem is damaged.

I had that issue a couple of times and it was that the SD card was actually corrupted or broken. So, a filesystem check might fix it or you will know that it needs to be replaced.

In Linux, you could check the filesystem with fsck.vfat. It works ok for a fat32 filesystem (not so for ntfs), so you could try that. Or use some other GUI utility like Gparted, Disks or whatever you like.

In Windows you should right click your SD drive (in my computer), select properties, the tools tab and check now. You can leave the auto fix errors checked if you don't mind losing files. (Disclaimer: I don't use windows, so the way to get to this might have changed in the last few versions).

If the card has corrupt sectors, the wisest thing to do is to format it (in vfat/fat32) and fill it with files (like mp3 or movies), then run another filesystem check. If it fails again, you should replace the SD.

ps1. Even if you don't have corrupt sectors, re-formatting the filesystem may fix the underlying issues. And if the problem persists, you should replace the SD.

ps2. Regarding ext4, it's not supported by default Android, you could use it with a partition if you have rooted the phone and Link2SD installed, but it's not recommended. ext4 has a journal system which would read &amp; write more often to the SD and diminish the cards life. ext2 is recommended instead (is the last ext filesystem without journal).

Update after re-reading:

Android 4 (kitkat), 5 (lollipop) and 6 (marshmallow) use all different methods to handle external storage (sd cards).

I didn't quite understand at first read that you were moving the SD card from one phone to the other.

As the access method to external storage changed between versions, this might be a permissions issue and you might be breaking some internal android settings.

You should try sharing the file with another peer to peer (email, telegram, syncthing, bluetooth or bittorrent sync) android utility or cloud based storage (dropbox, mega, wetransfer, etc.).