The following notes have a nice discussion of how feedback can make the response of a nonlinear static system more linear, i.e. reduce nonlinear distortion (at the expense of gain). https://web.stanford.edu/~boyd/ee102/fdbk-static.pdf
Is there similar reasoning for a nonlinear dynamic system?
For example, are there conditions under which the mapping from $u(\cdot)$ to $y(\cdot)$ in the closed loop system $\dot{y}=f(y,u-ky)$ is more linear than in the open loop system $\dot{y}=f(y,u)$?
Yes, there exists a method doing such a linearization also for dynamic nonlinear systems and it is called feedback linearization. However, they do not use a linear feedback as you stated in the question, but rather a nonlinear state feedback of the form $$ u(x) = \alpha(x) + \beta(x)\nu, $$ along with a change of variables $z = T(x)$ which render the system linear from $\nu(\cdot)$ to $y(\cdot)$. There exists the full state feedback linearization and the input-output linearization, where the system is only partly linearized to achieve only a linear input-output behaviour. The conditions for the existance of such a feedback law and the feedback law along with the change of variables $T(\cdot)$ involve the relative degree of the system and are detailed in Chapter 13 of Khalil: Nonlinear Systems. Prentice Hall, 2002.