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We prove that every quasiconformal mapping from the harmonic $\beta$-Blochspace between the unit ball and a spatial domain with $C^1$ boundary isglobally $\alpha$-H\"older continuous for $\alpha<1-\beta$, with the H\"oldercoefficient that does not depend neither on the mapping nor on $\beta$. Ananalogous result also holds for Lipschitz continuous, quasiconformal harmonicmappings for $\alpha <1$. This extends some results from the complex planeobtained by Warschawski in \cite{Warschawski} for conformal mappings and Kalajin \cite{Kalaj6} for quasiconformal harmonic mappings.

H$\mathbf{\"o}$lder continuity of QCH mappings from the unit ball to a domain with $C^1$ boundary