Is your domain clustered?
Are you using eis/HAFileAdapter instead of eis/FileAdapter connection factory?
The new files which the composite does not process, do they have same name as the old files which were processed?
There's a problem with deleting files from a file share
If I don't select the Archive & Delete functionality when configuring the adapter all files in the directory are processed correctly. However they do remain in the directory so there is a risk if some updates any of the files that they could be processed again.