Every non-erasing monoid morphism $\sigma: {\cal A}^* \to {\cal B}^*$ induces a measure transfer map $\sigma_X^{\cal M}: {\cal M}(X) \to {\cal M}(\sigma(X))$ between the measure cones ${\cal M}(X)$ and ${\cal M}(\sigma(X))$, associated to any subshift $X \subseteq {\cal A}^\mathbb Z$ and its image subshift $\sigma(X) \subseteq {\cal B}^\mathbb Z$ respectively. We define and study this map in detail and show that it is continuous, linear and functorial. It also turns out to be surjective. Furthermore, an efficient technique to compute the value of the transferred measure $\sigma_X^{\cal M}(\mu)$ on any cylinder $[w]$ (for $w \in {\cal B}^\mathbb Z$) is presented. Theorem: If a non-erasing morphism $\sigma: {\cal A}^* \to {\cal B}^*$ is recognizable in some subshift $X \subseteq {\cal A}^\mathbb Z$, then $\sigma^{\cal M}_X$ is bijective. The notion of a "recognizable" subshift is classical in symbolic dynamics, and due to its long history and various transformations and sharpening over time, it plays a central role in the theory. In order to prove the above theorem we show here: Proposition: A non-erasing morphism $\sigma: {\cal A}^* \to {\cal B}^*$ is recognizable in some subshift $X \subseteq {\cal A}^\mathbb Z$ if and only if (1) the induced map from shift-orbits of $X$ to shift-orbits of $\sigma(X)$ is injective, and (2) $\sigma$ preserves the shift-period of any periodic word in $X$.