Here is a modification of the example in my comment that avoids the mistake identified by Stefano. Let $Y$ be the projective cone in $\mathbb{P}^3$ over a smooth plane cubic curve $C\subset \mathbb{P}^2$. Let $\pi:X\to Y$ be the minimal desingularization.
Let $H\subset C$ be the restriction to $C$ of a hyperplane. Let $D\subset C$ be a degree $3$ effective divisor such that the divisor $H-D$ has finite order $\ell>1$ in the Picard group of $C$. Let $M\subset Y$, resp. $N \subset Y$, be the cone over $H$, resp. $D$. Thus, $\ell N$ is linearly equivalent to $\ell H$. The pullback of $H$ is the strict transform of $H$ plus the exceptional divisor $E$. Thus the pullback of $\ell H$ equals the strict transform plus $\ell E$. Therefore also $\ell N$ equals the sum of the strict transform plus $\ell E$. So the transform of $N$, as the OP defines it, is the sum of the strict transform of $N$ and $E$.
(Of course now you will claim that I am cheating, since I am using the fact that "division" is not uniquely-defined in the Picard group of $X$. Yeah, yeah, yeah ...)