The short answer is that the duty cycle cannot be increased because it's impossible to do so. Consider the DCM flyback case: -

[![enter image description here][1]][1]

The slopes of the rise and fall of transformer current are wholly dictated by \$V_{IN}\$ and \$V_{OUT}\$ for an inductor of a fixed value. This is also true in CCM but, CCM has nowhere to accommodate a change in duty cycle: -

[![enter image description here][2]][2]

The images above come from [my basic website](http://www.stades.co.uk/Flyback/flyback%20DCM2.html) and are for a flyback converter with a 1:1 transformer. I also have a waveform generator that you can use for testing the change from DCM into CCM as the load current increases. This is an example for a 1:1 transformer where the output voltage equals the input voltage (125 volts). DCM Operation is at 100 kHz with a 1000 Ω load: -

[![enter image description here][3]][3]

If I decrease the load to 800 Ω, operation is on the boundary between DCM/CCM: -

[![enter image description here][4]][4]

If I lower the load to 600 Ω, operation is clearly in CCM: -

[![enter image description here][5]][5]

The duty cycle cannot change because there is nowhere to accommodate that change hence, the whole primary and secondary waveform rises.

> *How do flyback converters operating in CCM regulate their output?*

It naturally regulates in CCM due to the way the slopes of primary and secondary currents link together.

> *What is the controller doing to maintain the output voltage if the
> duty cycle is constant like that?*

The controller doesn't need to do anything, it naturally happens once CCM is entered. 

If the charging phase of the switching cycle begins when there is still magnetic energy in the transformer core (left over from the transfer phase) then, the charging current has to commence at a positive value and not a zero amp value (as in the case of DCM). That cannot be avoided in CCM.


  [1]: https://i.sstatic.net/oTo1vVQA.png
  [2]: https://i.sstatic.net/JfOb4Ds2.png
  [3]: https://i.sstatic.net/QHzPcFnZ.png
  [4]: https://i.sstatic.net/rOlW7MkZ.png
  [5]: https://i.sstatic.net/fzGMc1S6.png