Can I transfer my non-B visa to a new passport at any immigration office in Thailand?

If you are British tell them you need the letter. VFS fobbed me off saying I didnt need it but immigration asked for it.

I am doubtful you are actually talking about a Non-B visa and it's far more likely that you mean an 'extension of stay' based on employment (look at the stamp in your passport)

You take both passports to the office that issued your extension and transfer the stamps.

With over 80 offices it's tough to say what exactly they will require but most require copies of data page of your old & new passport, copies of the Non-B visa and any extension stamps, your most recent entry stamp to thailand.

There is no 'standard form' that all the offices use just get any form they require when you're at that office.