How to design a state machine in face of non-blocking I/O?

I'm using Qt framework which has by default non-blocking I/O to develop an application navigating through several web pages (online stores) and carrying out different actions on these pages. I'm "mapping" specific web page to a state machine which I use to navigate through this page.
This state machine has these transitions;
Connect, LogIn, Query, LogOut, Disconnect
and these states;
Start, Connecting, Connected, LoggingIn, LoggedIn, Querying, QueryDone, LoggingOut, LoggedOut, Disconnecting, Disconnected
Transitions from *ing to *ed states (Connecting->Connected), are due to LoadFinished asynchronous network events received from network object when currently requested url is loaded. Transitions from *ed to *ing states (Connected->LoggingIn) are due to events send by me.
I want to be able to send several events (commands) to this machine (like Connect, LogIn, Query("productA"), Query("productB"), LogOut, LogIn, Query("productC"), LogOut, Disconnect) at once and have it process them. I don't want to block waiting for the machine to finish processing all events I sent to it. The problem is they have to be interleaved with the above mentioned network events informing machine about the url being downloaded. Without interleaving machine can't advance its state (and process my events) because advancing from *ing to *ed occurs only after receiving network type of event.

How can I achieve my design goal?

EDIT

1. The state machine I'm using has its own event loop and events are not queued in it so could be missed by machine if they come when the machine is busy.
2. Network I/O events are not posted directly to neither the state machine nor the event queue I'm using. They are posted to my code (handler) and I have to handle them. I can forward them as I wish but please have in mind remark no. 1.

3. Take a look at my answer to this question where I described my current design in details. The question is if and how can I improve this design by making it

   • More robust
   • Simpler

Sounds like you want the state machine to have an event queue. Queue up the events, start processing the first one, and when that completes pull the next event off the queue and start on that. So instead of the state machine being driven by the client code directly, it's driven by the queue.

This means that any logic which involves using the result of one transition in the next one has to be in the machine. For example, if the "login complete" page tells you where to go next. If that's not possible, then the event could perhaps include a callback which the machine can call, to return whatever it needs to know.