Visa Transfer to new Passport

Well here is my story, for US, don't know your country or city, and I didn't exit at Cambodia.
In January, even though my US passport had one year left, I will be moving to Thailand and traveling so I got a brand new 10 year passport at US Consulate Saigon. One reason is that it is faster there: only took 5 business days vs online warnings of 3 weeks in San Francisco. The officer at the Consulate said it will be no problem when I leave the country, just show the old passport with my valid visa stamp (I was on a 3 month extension) and the new passport. It is customary to identify an old passport as invalid by punching holes in it.

But I also married a Vietnamese in January, so I wanted to get a 5 year Visa Exemption Certificate (VEC). At both the clean Saigon Immigration office on Nguyen Thi Minh Kai, and the ugly Vietnam Immigration office in Nguyen Trai, I was told the first step is I had to get the visa stamp moved to the new passport. But neither place wanted to do it. They said, go to the other office. grrr! There are a bunch of 'visa helpers' hanging around inside Ng Trai, they will approach you in there, they offered to get it done for $35 I think. I asked my regular visa guy, he said $50. Well, It turned out that getting a VEC was going to require leaving and reentering the country anyway (crazy) so I decided to forget about it, and just deal with two passports at the airport when I take a trip to the US, and apply for the VEC at the San Francisco Vietnam consulate where things should be more orderly, and in English.

Two days ago I passed through immigration at Tan Son Nhat airport on my way to SFO. The immigration guy looked at my 2 passports and stood up and said a tieng viet equivalent to 'Jesus h christ!' and called over another guy. Guy #2 also shook his head and grumbled, and then they both escorted me to the side area for troublemakers and gave my passports to guy #3 in a booth. I had to sit down for awhile in a row of seats. Then guy #3 sticks up his head and in perfect English asked where I got the new passport. I said 'US Consulate Saigon'. One minute later he came over and very politely gave me the passports and said 'thank you you can go'. The extension stamp was copied into the new passport. No charge.

Yes you can transfer your visa to new PP.
When new passport issued, they will cancel (by punch hole) the old one. Remember, cancellation is your old PP, not the affixed visa!
Normally you can keep and show when you exit. Just keep the old PP until you transfer your visa to new PP or until you renew your current visa which they will affix in the new PP.

Being that I have a business Visa without sponsorship (self employed tutor), was basically told by immigration today that my Visa cannot be transferred to the new passport.

Yet I have been instructed by my Visa rep to visit airport immigration and see what they can do. To my knowledge, border offices will not transfer the visa.

Will report back with the airport results and see if it is a viable option.

I got my new passport and had my VEC in my old cancelled passport and they would not except the visa any longer. Had to get a new one. But I too was told the story by the US Consulate that my visa could be transferred to the new. Of course, it is not their visa so why they make decisions on another countries visa I do not know. But in my case they were wrong. We are in VN, things change daily and according to the person you talk to. Go to the next window and thing are likely to be different.