I would like to copy and move questions across multiple question sets. Is there any way to do so?

Yes, there is. Apart from moving and copying questions within the same question set, you can also do so across the sets.

1 Possibilities of moving questions.

You may want to use the option “Move/copy into a new question set” to create a new question set from the questions contained in another set created before. To do so, click on the aforementioned button, i.e. “Move/copy into a new question set”, and select those of the questions that you want to move/copy (you should specify which of the two operations you want the System to perform). Provided you decide on the former, the selected questions will be removed from the source question set. You will also be prompted to enter a name for the target set and specify the folder you want it to be stored in. To complete the moving/copying operation, please click on “OK”.

PozorUsing the aforementioned technique, you cannot move questions from a question set A to that of B created before! That is, the operation always creates a brand new question set (the target one)!
Another option you might find useful is that of “Move/copy from another question set”, which allows you to load a question set with all the questions making up another set. To do so, click on the “Move/copy from another question set” button and specify the question behind which you want the other set to be pasted. Provided you check the remove the source file option, the source question set will be removed. To complete the operation, click on “OK”.