I believe that this could be because of the default settings in Transmission. By default, Transmission is configured to "hit & run". Fear not though as there is a rather easy way to fix this little problem. I too use Transmission and fell victim to it. To fix this issue, please do the following:
Open Transmission
Click the "Edit" menu, then click in "Preferences"
Click the "Seeding" tab
Uncheck both of the checkboxes ("Stop seeding at ratio" and "Stop seeding if idle for N minutes")
Click the "Close" button
Start the torrents that otherwise say "Finished" or "Paused".
Once you have done this, your desired torrents will continue to seed regardless. You can also change the setting for each individual torrent after it is added by right clicking the torrent, then going to the "Options" tab and looking for the "Seeding" options. By default, Transmission will assume the global settings when a torrent is added, however it is possible to change these settings for an individual torrent if you so choose.