My current Retirement Stamp was obtain in Chiang Mai where I was living at the time. My rental contract ended on 15 January 2016 and I went to UK for a couple of weeks. I am going to the UK again in March and whilst I am there will apply for a new passport. When I return I will move elsewhere in Thailand. If I submit a change of address to the Immigration of wherever, will they, with the support of the Consulate letter, transfer the stamp or do I have to go back to Chiang Mai.

When you return, walk in to your new Immigration Office and ask them. If they say no ( highly likely), then go to Chiang Mai. If you are posting because you have the option of moving first and travelling second,that would eliminate your problem.

Got my original stamp from the Thai Embassy in Los Angeles. Had my stamp transferred to new passport in Korat where I now live with no problems.

Assuming that the OP (1) is in receipt of the UK State Pension and (2) doesn't have much longer than 2-3 months remaining of his current extension (which may well both be big "if's", I know), I wonder whether the simplest course of action for him might be to obtain a fresh single-entry non-O visa in his new passport while he is in the UK? He would then be stamped in for 90 days upon his return to Thailand and he could then apply for a fresh extension of stay at his new immigration office within 30 days of the expiry date (or maybe as early as 45 days beforehand in the case of some offices).

But, regardless of the course of action which he follows, he will almost certainly need to submit completed forms TM28 and TM30 at his new immigration office before they are prepared to do anything in any event.

When does your current 1 year extension of stay ("Retirement stamp") expire?