When we were travelling through the USA we definitely had instances of dd putting her foot down and not wanting to come with us. The only thing that worked was to emphasize the fact we knew she didnt want to go but we were a team, and we had to work together. I also say things like 'mummy keeps you safe so you need to stay with me' and 'mummy helps you when you want a drink/go to the toilet/need something to eat, so you need to help mummy and come when I ask'.
Curious did you ever get the "but I can do it myself, I don't need your help" response to this? Any suggestion by me that DD needs me is not welcomed by DD at all. Sometimes this gets me quite stuck but I suppose her believing in herself is a good thing really.