Let $f:\mathbb{R}^n\to \mathbb{R}^m$ be a smooth mapping.
Let $X$ be a submanifold of $J^r(\mathbb{R}^n, \mathbb{R}^m)$.
Supposing that the submanifold has the following property :
if $j^r\sigma(p)\in X$, then $j^r(g\circ \sigma \circ h^{-1})(h(p))\in X$ for any diffeomorphisms $g:\mathbb{R}^m\to \mathbb{R}^m$ and $h:\mathbb{R}^n\to \mathbb{R}^n$.
Suppose that $j^rf:\mathbb{R}^n\to J^r(\mathbb{R}^n, \mathbb{R}^m)$ is transversal to the submanifold $X$.
How can we prove $j^r(g\circ f \circ h^{-1}):\mathbb{R}^n\to J^r(\mathbb{R}^n, \mathbb{R}^m)$ is also transverse to $X$?
(Here, note $g$ and $h$ are diffeomorphisms)
In the case $n=m=1$ and $r=1$, it is easy. In the situation, for example, put $X=\mathbb{R}\times \mathbb{R}\times 0$. If $j^1f(x)\in X$, by the asuumption $f$ is transversal to $X$, it follows that $df_x\not=0$. Becasuse $g$ and $h$ are diffeomorphisms, it is easy that $d(g\circ f \circ h^{-1})_{h(x)}\not=0$. So $j^1(g\circ f \circ h^{-1}):\mathbb{R}^n\to J^1(\mathbb{R}^n, \mathbb{R}^m)$ is also transverse to $X$.
But I can't show the general cases $(n,m\geq 2,r\geq2)$.