@Benjamin ********
You present both passports. Immigration won't transfer visas. They will only transfer extensions of stay, obtained in country and you would do that through your assigned immigration office only.
If your passport expires with a valid DTV in it, you carry the old passport along with the new passport whenever you enter and exit Thailand.
I've done that before with a non-O multiple entry visa. The only time I got my visa (actually extension of stay) transferred was when I had an in country non-O 1 year extension, which was then transferred to my new passport. That took up 3 pages in my passport! They copied all the previous extensions as well.