You should use Core.getFileDocumentContent instead of trying to access filedocument files directly by path.
The FTP client does not support using an InputStream sadly enough. Roeland reply got me thinking:
You should use Core.getFileDocumentContent instead of trying to access filedocument files directly by path.
The solution I came up with is the following: I use the InputStream from getFileDocumentContent() to generate a File in the Mendix Temporary folder based on the name of the FileDocument in Mendix. After generating the File, I send it to the FTP location and afterwards, I delete the generated Temporary File.
Thanks Roeland.