I was in a similar position, being an experienced Internal Consultant/BA/PM with 10+ years of project experience. I've recently joined a large consultancy however, and had offers from 2 others, so it's certainly possible.
It does seem that industry expertise is being sought by the big firms nowadays. Presumably it is easier to train someone in basic consultancy and sales techniques than it is to educate someone in the equivalent of 10yrs+ of industry expertise.
The main difference is the increased focus on sales in the "external" world. Obviously as an Internal you will tend to get allocated work and as an External you will have to help go out and win it.
To this end, you will need to emphasise in your CV and subsequent interviews any "sales & client facing" experience you had in your role as well as your project execution experience. Transferable skills, such as involvement you have had in making pitches, presentations, change management communications etc will help show that you have the potential to win new business as well as deliver on exisiting business.
Also, if you have worked with external consultants on your projects, have a think of work they did that you were impressed with, as well as those areas that could be improved. It will help inform your interview if you can offer insight as to where clients may get frustrated by their external consultants, and where you can add value by avoiding that pitfall.
One other thing to bear in mind is that you may have to take a side step or small backward step in terms of your relative seniority. You may be director level equivalent in your company now, but are likely to be no more than a senior manager as a consultant, at least whilst you learn the ropes.
Good luck with it all.