IUD and Menstrual Cups

Isadora
So, I've had my IUD(copper) removed because it moved positions in my uterus and got another reinserted. For both, I have been used the menstrual cup. I was just wondering what experiences you guys have had with these two. 
I have made my own research, and menstrual cups do not move the IUD, but I have heard of occasions, so has this happened to anyone? Should menstrual cups and IUDs be used together? 
I will be making a Planned Parenthood appointment soon to ask these questions and possibly change my birth control, let me know what you guys think.