Yes - the moves must connect seamlessly. I can see well why someone might call it simply "ayumi".
See my previously posted form: http://www.youtube.com/watch?v=B56GG6bCdVU
at points 0:18 to 0:20 and 0:23 to 0:24 and note the diagonal step followed by the "irimi" - all flowing one into another. Without the flow, it simply doesn't work
I filmed the applicatons after class yesterday: http://www.youtube.com/watch?v=Cm2f66OkUOM
at 0:47 onwwards. It's bit messier than I would have wanted, but Jed (my attacker) isn't very compliant (which is good!).