Mine didn't work that way when I went from the 6 to the 6+. I was told that even though it was the same size SIM card there was a difference and wouldn't allow apple pay to work.
I spoke to a customer rep the other day because I'm doing the same thing, upgraded on my wife's line for the 7 and giving her my 6+. Customer rep said no problem and dropped a new SIM card in the mail to me for the 6+. They said before I turn the 7 on for the first time call them and they would associate the phone and SIM to my number. After that put the new SIM they sent me into my old 6+ and it would already be setup for her number.
Maybe this is necessary, maybe not but doesn't seem like a big deal and didn't cost anything for the SIM.