Since you can staff LibraryH3lp using any XMPP/Jabber client, you can staff your service from a mobile device if you so desire.
The staffing assignments dialog is something unique to LibraryH3lp, so you won't see an option to change staffing assignments with mobile XMPP./Jabber clients. The current settings for your LibraryH3lp staffing assignments will be used automatically. If you need to make changes to your staffing assignments, you'll need to briefly log into the webclient for staffing to do so. Alternatively, administrators can make changes to these staffing assignments for you via the users management page in the dashboard.
To access LibraryH3lp-specific features like transfer, tag for followup, etc on mobile, you'll use the chat management link that appears with every LibraryH3lp chat over a queue.
For Android, the Xabber app is very nice and recommended.
For iOS, AstraChat is probably the best option currently (year 2020).
If you are certain you have set up the app correctly and your login details are correct, and you get an error that the app cannot reach the server, it is likely that your XMPP port 5222 is blocked, either in a local firewall or further upstream, such as your ISP. These apps require port 5222 to be open in order to work.
Operators can seamlessly switch between desktop computers and mobile devices if they need -- even during a chat. For example let's say that an operator answers a chat on her mobile device, then decides she wants to continue the chat on her desktop computer with the webclient in Chrome. First, she would need to ensure she is logged into the webclient in Chrome and on Available status. Then she would switch to Away status in the mobile device chat client. The next message from the guest would appear in the webclient on her desktop computer instead of on her mobile device. And the opposite scenario -- taking the chat from webclient to mobile -- works similarly. The basic process is to go to Available status on the device where you want to chat and then go to Away or Offline status on the device/computer where the chat was originally answered.
While your mobile web browser might work in a pinch or for short periods of time, it is not ideal. You really need good push (background) notifications for new chat messages, and that requires a mobile chat app. Push notifications will allow you to do things like allow the device to go idle, lock the device, make another app active, etc, while still receiving reliable alerts of new chat messages. Test your app's notifications thoroughly.
You will also have a difficult time answering questions from your guests using a web browser on a mobile device, because if you need to bring another mobile browser window to the front in the process of answering the question, you might disrupt your network connection to the chat server.
Note that all of these settings probably will not be available in all mobile clients. This is a general listing of possible options across numerous clients.
If available in your app, you'll want to make sure that you have push notifications turned on for any mobile client you install. That way you'll be notified of incoming chats from guests. These clients will have have various settings you can configure for push notifications, including sounds, alerts, and badges. You might have to check in the app's settings AND in your device's general settings for Notifications (especially for iPhone and iPad).
Things to check when trying out a mobile client include push notifications (look carefully through all the settings both in the app and in the device's settings), how to change status, how to logout, and receipt of the initial three chat messages when receiving a LibraryH3lp chat on a queue (including the chat management link).
In our last rounds of testing, we found that Trillian had trouble with push notifications and IM+ was not able to connect to our XMPP server.