Canadian traveling to US and/or Mexico on British Passport

Hi. I have googled this question & thought you may have some advice.
I am a Canadian citizen/resident holding a Cdn passport who is planning to travel with a Canadian citizen resident with an expired British passport.
If they renew their expired British passport, are there any airline/customs & immigration issues to be aware of for a Canadian holding a British passport traveling from Canada to either a US or Mexican destination?
Thanks in advance.

Yeah, they'll have to use the visa waiver programme and register with ESTA: http://www.cbp.gov/xp/cgov/travel/id_visa/esta/

If they're Canadian and in Canada, I would have thought it would make more sense to get a Canadian passport than renew a UK one. They'll need a Canadian citizenship certificate or Canadian birth certificate to get back in otherwise in addition to the passport.

Zackly. A British citizen would have to enter the US on the VWP, and is only allowed to stay up to 90 days, a Canadian can enter and stay up to 180 days.

No idea about Mexico, I've never been there so I have no idea what the visa sitch is there at all.