Is an FIR filters 'transition width' frequency dependent?
You're right: the required filter order is approximately inversely proportional to the desired transition bandwidth $\Delta\omega$, regardless of the cut-off frequency. This is reflected in the empirical formulas for estimating the required filter orders for the Kaiser window design method as well as for the Parks McClellan equiripple design of low pass filters:

$$\begin{align}M&amp;=\frac{-20\log_{10}\delta-8}{2.285\Delta\omega}&amp;\qquad\text{(Kaiser Window)}\tag{7.63}\\{}\\
M&amp;=\frac{-10\log_{10}(\delta_1\delta_2)-13}{2.324\Delta\omega}&amp;\qquad\text{(Parks McClellan)}\tag{7.104}\end{align}$$

where $\delta$ is the maximum allowable approximation error (Kaiser), and $\delta_1$ and $\delta_2$ are the maximum pass band and stop band errors, respectively (Parks McClellan).

The equation numbers above are taken from the book Discrete-Time Signal Processing by Oppenheim, Schafer, Buck, 2nd edition.