The problem is: Windows maintains a 'dirty' bit for each simple volume that determines whether CHKDSK will be run on next boot for fixed media, or if the 'Do you want to scan and fix Removable Disk ...' dialog will pop up when removable media is mounted. Unfortunately the 'dirty' bit is MS-centric and poorly documented so Linux tends to ignore it on FAT16/FAT32 filesystems, and some operations can 'smash' it leaving the filesystem marked as 'dirty'. The situation with NTFS volumes is a bit different as if a 'dirty' volume is detected, Linux will mount the volume read-only.
Therefore on FAT16/FAT32 volumes on removable media that have been written to on a Linux device there is a high probability of getting a bogus warning that the volume is damaged. If you want to play safe, reject the warning, back up the data on the volume, then (safely) remove and reinsert and only then allow Windows to scan and fix the volume. If you are lucky, and careful to always use 'safely remove ...' on the Windows side, the cleaned 'dirty' bit may stick, avoiding future warnings.