I just tried moving a single file in both Astro File Manager and EStrongs File Explorer and both worked fine.
In Astro, long press on the file and select Move. Then navigate to the folder you want to move the file to, press Edit in the toolbar or Menu > Edit, and then select Paste.
In Estrongs, long press on the file and select Cut. Then navigate to the folder you want to move the file to, press Menu > Operations > Paste.
Based on the speed of the operations, it appears that Astro may be actually copying the file and then deleting the original, and that EStrongs is actually moving the file (changing the allocation table without actually moving the data).
Be aware that moving files from the memory card to a network drive (or vice versa) may be problematic or impossible, and that you may need to copy and the manually delete the original.