Cyclotomic multiple zeta values (CMZV), are an interesting first bunch of examples of periods and a fruitful recent approach is to look at their motivic version (MCMZV), which are motivic periods of the fundamental groupoid of ℙ_1 ∖ {0, μN, ∞}. Notably, MCMZV have a Hopf comodule structure, dual of the action of the motivic Galois group on these specific motivic periods; the explicit combinatorial formula of the coaction (Goncharov, Brown) enables, via the period map (isomorphism under Grothendieck’s period conjecture), to deduce results on CMZV. We will here highlight how to apply some Galois descents ideas to the study of these motivic periods and look at how periods of the fundamental groupoid of ℙ_1 ∖ {0, μN', ∞} are embedded into periods of π1(ℙ_1 ∖ {0, μN, ∞}), when N′ | N, via a few examples.