ITN - EID: In an EID with an academic and an non-academic partner (and other partners in other countries) in the same country, does the ESR have to change countries for his/her stay at the non-academic institution?

Yes, this is technically eligible. But as the international dimension is weaker than it might be in competitive proposals where the ESR goes for his/her stay to a non-academic partner in yet another country, it is not recommendable.