Now with the latest release of Salesforce Mobile app version, now the tabs added in desktop version automatically get added into the mobile app also.
Just enable 'New Salesforce Mobile App' user permission to certain/all users & then you are good to go.
Refer this link: here
I too needed this feature & by doing so, it worked.
Emmanuel, as you've confirmed... if you want to use the Lightning Page framework for mobile, you must enable the Mobile on Lightning Permission: https://releasenotes.docs.salesforce.com/en-us/summer19/release-notes/rn_general_new_mobile_app_pilot.htm.
In case this link goes away... one ultimately needs to add the "Lightning on Mobile" System Permission to ...
I found that the answer to this was to use a redirect with the EXACT correct link. I had to use the old links like /servlet/networks/switch?networkId=[yournetworkidhere] instead of trying to use the URL of the community itself for the redirect.