In the article, the boundedness of vector-valued sublinear operators in grand variable Herz-Morrey spaces MK˙λ,p(⋅)η(⋅),q),θ(Rn)
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are obtained. Then grand variable Herz-Morrey type Besov and Triebel-Lizorkin spaces are defined. We will also prove the equivalent quasi-norms by Peetre's maximal operators in these spaces.