Originally a blog article (How to Throw?) her at last is a home for it.
First we have to get to the centre of mass of uke, then we can connect / take it, then its time to wrench on it a bit and try to get it out past ukes feet. Here are some steps to help understand the problem. Understanding is the first step in being able to do.
Getting Connected
If life was simple, at least in aikido anyway, ukes body would be a giant rigid mass we could, with care, move around easily (see Aikiphysics - Fridge Aikido). So somehow we have to turn ukes body into something rigid. Fortunately by stretching joints, compressing them and tightening them with the array of available tools in the aikido belt this is certainly possible. Take ikkyo for eample you can stretch out your partners arm, compress it into their body and rotate the joint a little to stop pesky things like elbow bends and have a nice solid connection to their centre (remember you have to do the same to ukes centre line/ spine as well)
Freezing the base
With a solid connection in place, now we are ready to shift Ukes Centre of Mass (CoM) out past the base of their feet and once achieved uke will start to fall over. We can give this a helping hand by applying our own power to the task (See Aiki Physics Generating Power, Transferring Power). The only problem is that ukes don't always know when to fall over and do things like taking steps to stop it happening. So the answer is to freeze the base by cutting through the centre to the ground or stretching up high. Traditionally this might be referred to as grounding.
Where to from here
Having frozen ukes feet to the ground its time to move it out from their base. Pushing into uke puts the weight on the back foot and the front foot is free to be lifted up for a step..or worse a front kick. Pull uke toward you and uke is free to move the back foot for stepping and maybe other nasty tricks. Along this line (in blue in the image) is the most extreme case, other angles can also allow uke to lift their feet and have mobility.
The answer then is to move their centre in such a direction that the weight is evenly directed onto both feet (along the red line). Traditionally we are heading for Shikaku behind them or Omote in front of them. If you take the centre a long way out you have a bigger throw or just move a little bit to have them fall more gently. You can also send your power out, up or down depending on the technique (sometimes all three are possible)
Once these ideas are understood they can be a helpful way of asking yourself a number of questions that you now have the tools to answer to improve your aikido
'Am I standing in the correct place at each stage of the technique?'
'Am I connected to uke appropriately at each stage in the technique so that they cannot move their feet until I am ready for them to take ukemi?'
'Can I do this with less power and still make it work?'
'Can uke exploit my weak points with these ideas?'
These ideas are generalisations of lots of stuff but i hope of some use. So next time you are bored in class (not my classes i hope) ask yourself these questions, there is a few years work in them I suspect.
Addendum
Interestingly in the daito ryu work shown by Okajima sensei it seems the centre is very subtly moved to just outside the stable region such that Uke is unaware of their in stability. Nage can aid this feeling of stability by proving support for Uke. Once released Uke finds themself falling - sometimes with a perplexed expression. Still chewing this one over