I would like to know how much of the recent results on the MMP (due to Hacon, McKernan, Birkar, Cascini, Siu,...) which are usually only stated for varieties over the complex numbers, extend to varieties over arbitrary fields of characteristic zero or to the equivariant case.

I assume that the basic finite generation results hold for any such field, by base extending to an algebraic closure, so I would guess that most results should extend without too much difficulty. The particular questions that I am really interested in are:

1) Given a smooth projective rationally connected variety X over a field k of charaterisitic zero, can we perform a finite sequence of divisorial contractions and flips to obtain a Mori fibre space?

and

2) The equivariant version of 1) for the action of a finite group on X.