Letkbe a commutative ringGa finite groupRandSfullyG-gradedk-algebras. In this paper we investigate Morita equivalences, derived equivalences and stable equivalences of Morita type betweenRandS, which are induced byG-gradedR, 5-bimodules or complexes ofG-graded bimodules. Such equivalences occur naturally in the case of group algebras in certain reduction steps for Broue's conjecture, and we show how they can be lifted from equivalences betweenR1andS1.