Let $\phi:\mathbb{A}^n \to \mathbb{A}^m$ be an algebraic map. Denoting $X = \mathbb{A}^n$ and $Y = \mathbb{A}^m$ we define $D_{X \to Y} = k[X] \otimes_{k[Y]}D_Y$. So its elements are sums of elements of the form $f \partial $ where $f$ is a polynomial on $k^n$ and $\partial$ is a product of differentials with repspect to the functions $\phi_1,...,\phi_m$ (is that a good way to think about $D_{X \to Y}$?)
We define the pullback of a left $D_Y$-module by $\phi^0 (N) = D_{X \to Y} \otimes_{D_Y} N$ and the pushforward of a right $D_X$-module by $\phi_0(M) = M\otimes_{D_X}D_{X\to Y}$.
I can then think of its elements as sums of formal products $hf$ where $h$ is a polynomial on $k^n$ and $f \in N$, which makes sense to me when we take for example $N$ to be the module of differential functions on $Y$, because I can see how these are differential functions on $X$.
However I'm not sure I can intuitively understand the pushforward. It seems to me that
$$M \otimes_{D_X}k[X] = M/(M\partial_{x_1} + ... + M\partial_{x_n})$$
(under the isomorphism $\xi \otimes 1 \leftrightarrow \xi$)
Which would mean, denoting $M\partial = M\partial_{x_1} + ... + M\partial_{x_n}$, that $$\phi_0 (M) = M/M\partial \otimes_{k[Y]} D_Y$$
So.. its elements are sums of elements of the form $\xi d$ for $\xi \in M/M\partial$ and $d$ a differential operator on $Y$. This kind of makes sense to me in the context of, say, $M$ being the module of distributions on $X$. But it would make a lot more sense if it was $M \otimes_{k[Y]} D_Y$. I don't understand where the quotient by $M\partial$ comes from.
Where does it come form? Also, am I thinking about things here correctly? I am trying to build intuition, and if there is a mistake in my intuition or maybe another example of a $D$-module you usually consider when you try to make sense of definitions, I would like to know :)