Thank you for your question.
At this moment in Adobe Muse you can build adaptive layouts for Desktop, Tablet and Mobile devises separately. This means you have an option to choose which layout you want to create before starting a new project. For example, you can create a website for Desktops only or Desktops/Mobiles or maybe Desktops/Tablets.
In this particular case you need to customize the mobile version of the template as well.
Adaptive layouts allow you to create even a new designs for the each layout.
When your site is ready to be published on the hosting, all you need is to export site and the Muse will export all ready layouts for you.
P.S. Adobe Muse team announced that Muse is going to be responsive soon. That means you will be able to choose what kind of website you would like to create, either Responsive or Adaptive.
I hope this helps.