Isle of Wight
At the start of my turn, after adding 3 knights to my castle, I have 6 knights in my castle, 4 defending a fort, 3 defending a marshlands, and an unoccupied ruins next to me. I want to attack the ruins, so I put 1 knight there.
Can I, at the same time, use the left over 5 knights in my castle to move into the fort and marshlands, bringing their numbers up to 6?
The example in the rulebook only mentions removing knights from an area.
City of Bristol
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To answer your question: yes ...yes you can.
Re: Advancing = redistribution?
In the first edition, it is written that you redistribute your knights while you make an advance. In this case, you can advance any number of knights into an adjacent territory you haven't conquered, up to its limit, while you redistribute your remaining knights among the territories you have conquered (with a line of supply) in any way you choose.
On the other hand, for the second edition, we have defined redistributing and advancing separately, and much more clearly. First you redistribute you knights among the territories you control (and that have a line of supply) in any way you choose. It is at this point you redistribute a certain number of knights into your castle to prepare for your advance. You then make your advance by moving those knights into any territory adjacent to a territory you control (that has a line of supply), up to its capacity.
This is clearly explained in our latest edition of the rule book, found here.