Rainbow Icing Troubles - Help Please!

Hi, I've tried making rainbow icing a few times and I always have the same happen. Can someone tell me where I'm going wrong?

So I make my icing, I separate it into 6 or 7 portions, I mix in the colour, so then I have my coloured icing ready to go. I then put each colour into it's own piping bag, and pipe out a stripe of each colour side by side onto some cling film/saran wrap. I then roll up the cling film rainbow and cut the end and then insert the whole thing into a new piping bag with whatever tip I'm using.

The problem is that after a couple of beautiful rainbow cupcakes, the different icing colours start to mix together and I end up with a brown/green mix that's not beautiful.

What am I doing wrong? How do I keep the colours from mixing together? Help please!

ETA: I'm using buttercream