I recently copied a column of text and pasted it into a new document.
In the the original the font was Minion Pro but in the pasted version the font had changed to Times new Roman.
Is there something I can do to prevent this behaviour?
This is likely due to paragraph styles with the same name in each file, but different properties. For example, if you make a new document, the default paragraph style is called Basic Paragraph, and for me, the default font is Minion Pro. If you were to make a bit of text in that document and then paste it into a different document that defined Basic Paragraph with a different font (like Times), the font would change to Times when you paste into the other document. That's a good reason to not use Basic Paragraph, and not to base new paragraph styles on Basic Paragraph.
One option is to change the name of the style before you paste it (if the cause of the problem is that the same paragraph style name is used in both documents) so that the new style name will be added to the new document, and the formatting will stay the same. Another way is to break the link to the style in the first document before copying, but you get more control when styles are applied, so changing the name is a better option (in my opinion). You can rename styles, but Basic Paragraph seems to be the exception, so creating a new style and applying it before pasting is the best way to go (also my opinion).