Let $k$ be a positive integer. A subset $S$ of $V(G)$ in a graph $G$ is a $k$-tuple total dominating set of $G$ if every vertex of $G$ has at least $k$ neighbors in $S$. The $k$-tuple total domination number $gamma _{times k,t}(G)$ of $G$ is the minimum cardinality of a $k$-tuple total dominating set of $G$. In this paper for a given graph $G$ with minimum degree at least $k$, we find some sharp lower and upper bounds on the $k$-tuple total domination number of the $m$ -Mycieleskian graph $mu _{m}(G)$ of $G$ in terms on $k$ and $gamma _{times k,t}(G)$. Specially we give the sharp bounds $gamma _{times k,t}(G)+1$ and $gamma _{times k,t}(G)+k$ for $gamma _{times k,t}(mu _1(G))$, and characterize graphs with $gamma _{times k,t}(mu _1(G))=gamma _{times k,t}(G)+1$.