It's for sure that you don't need bushes to turn between centres, you just need to keep measuring as you go so that you don't turn the blank under size, If you are going to use bushes then you can use them as they come, but you will find your centres start to get damaged unless the have a hardened tip, better that they have a 60 degree chamfer to them to match the centres, they will run much better that way, but just try them as is and see if they do what you want.
If you have a mandrel that uses collets like the Axminster deluxe mandrel then you can get different sized collets for that and depending how big the bushes are you can set them in and use a centre drill for the 60 degree entrance.
Peter