Kokyu-nage is a loose term. I would consider them as uki-waza (aka floating technique) which requires shite to extend uke to the point of unbalance, then add a little push at the correct direction to effect the throw/projection.
Take Shomen Uchu. You can step through and turn, guiding the attacking arm down for a kokyu throw. If you mis time it or they recover you can maintain connection and draw them through for a shiho nage.
The same can be said if you draw them out and subsequently stick your leg out to impede uke's movement, you'll probably get O-Guruma or Tai-Otoshi.