Yes. We have secondees from some of our main clients. We had a guy seconded from the Cabinet Office for about a year until this summer.
These things are always (to my knowledge) based on relationship-building i.e. the consultancy only takes on the secondee if they know they can trust the secondee and their organisation with any commercially sensitive information.
Secondments with consultancies are organised corporately and are not open to independent, external applications from individuals (hence why you don't see them advertised).
There's no immediate benefit to a consultancy in having someone for a short period (generally a year) who is not necessarily qualified to be billed as a consultant but can't be given more junior roles either. The benefit comes from the secondee giving the consultancy work once they return to their normal employer.