When you experiencing that downloaded files have a wrong extension in Firefox (.part) enable either Permalinks (SEO friendly URLs) in WordPress Settings or enable “Always Force Download” in WP-Filebase Download settings.

FYI: one of these options must be enabled, because otherwise the browser gets no information about the file name, only the MIME type is sent in HTTP headers. Chrome & Safari seem to handle it without a Content-Disposition Header (i.e. Chrome names files “Download.pdf” if the content type is application/pdf). Firefox does not care about the content type and just gives the file the extension .part.