At the moment, the selected note moves by its rhythmic duration, so this half note has moved forward a half note. You will have experience the above situation because you have Insert switched on, meaning that the low E will not overwrite any of the G. Instead, what is happening is that those two half note long portions are swapping places so that not music is lost. In my attachment, I’ve tried to show how the eighth note rest and the first dotted quarter note of the half note G swap places with the E.
It just gives you extra flexibility when moving notes around, being able to choose whether they should overwrite other music or not. Finer control over how the notes are moved will come in an update.
I suspect I am vastly confused by the implications (more than the function) of the Input option–when to use it and when to turn it off, as it can move different players out of sync if used at the wrong time.
In your original example you asked Dorico to move the half note one half note to the right. So it did. Because the new start location was part-way through an existing note, Dorico split it for you.
This makes me laugh. In INSERT mode Dorico is supposed to push the existing notes on its right side (in this case the G note) forward in time and not to “split” them.
PS: Sorry, I am not laughing about you. I am laughing about the above splitting push result being a “great feature”.