We study local G-shtukas with level structure over a base scheme whose Newton
polygons are constant on the base. We show that after a finite base change and
after passing to an \'etale covering, such a local G-shtuka is isogenous to a
completely slope divisible one, generalizing corresponding results for
p-divisible groups by Oort and Zink. As an application we establish a product
structure up to finite morphism on the closed Newton stratum of the universal
deformation of a local G-shtuka, similarly to Oort's foliations for p-divisible
groups and abelian varieties.