Re: Ai waza
I think by your description, the technique you are after is called Aiki Otoshi or in some circles Aiki nage.
This technique can be performed from different attacks, however the usual attack to learn the technique (well this is how I was taught it at least) is from Ai hanmi or Shomen uchi.
Enter irimi to the side of uke lower your knees which in turn drops your weight and unbalances uke.
with your hips firmly pressed against uke`s, and your arms wrapped around their waist straighten your legs and twist your hips to the rear, throwing to the rear over your hips.
hope this helps :-)