  We define etale cohomology of the moduli spaces of mixed characteristic local
shtukas so that it gives smooth representations including the case where the
relevant elements of the Kottwitz set are both non-basic. Then we relate the
etale cohomology of different moduli spaces of mixed characteristic local
shtukas using convolution morphisms, duality morphisms and twist morphisms. As
an application, we show the Kottwitz conjecture in some new cases including the
cases for all inner forms of $\mathrm{GL}_3$ and minuscule cocharacters. We
study also some non-minuscule cases and show that the Kottwitz conjecture is
true for any inner form of $\mathrm{GL}_2$ and any cocharacter if the Langlands
parameter is cuspidal. On the other hand, we show that the Kottwitz conjecture
does not hold as it is in non-minuscule cases if the Langlands parameter is not
cuspidal. Further, we show that a generalization of the Harris--Viehmann
conjecture for the moduli spaces of mixed characteristic local shtukas does not
hold in Hodge--Newton irreducible cases.
