When you are using the Alternate Pages approach, styling desktop and mobile pages differently, you will likely also want to change the appearance of the Layout Elements - including the navigation.

This is, essentially, logically no different than any other styling, but it is so significant it is worth describing separately.

The method here is to have two sets of Layout Elements configured, one for desktop, and one for mobile, and then in the BE / Settings choose which ones should be shown on the respective pages. As with all Behavior Editor settings, to avoid repetition, make the settings as high up the site tree as you can.