Let $v$ be a constant vector field $v=\sum_{i=1}^n c_i\dfrac{\partial}{\partial x_i}$, and let $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a bijective linear map. What is the vector field $f_*v$?
So, the vector field $v$ maps each point $(x_1,\ldots,x_n)$ to the vector $(x_1,\ldots,x_n,c_1,\ldots,c_n)\in T_p\mathbb{R}^n$. Hence, the push-forward vector field $f_*v$ must map $f(x_1,\ldots,x_n)$ to $$(f(x_1,\ldots,x_n),Df(x_1,\ldots,x_n)\cdot (c_1,\ldots,c_n))\in T_p\mathbb{R}^n$$
What else can we say about it?
The total differential $Df$ of a linear map $f$ is just the map itself. This follows directly from the definition of the derivative as a linear map given e.g. in this article.