Basically comes down to this: forced CCM is 100% the same control response over all operating conditions, period; whereas, when DCM occurs, the modulator gain at negative command (i.e., negative output current or falling Vout) drops to zero. And so the dominant pole due to the filter cap shifts towards zero. When it falls near the controller's dominant pole, they split and diverge (increased overshoot), and further still, they're pushed into the right half-plane (oscillation).
The choice then is: do you tune the controller for very low bandwidth, to keep stability, arbitrarily deep into DCM? (You can't go on forever, because when load current goes to zero, the capacitor pole equals zero: an ideal integrator.)
Or do you tune it for CCM operation (prioritizing transient response), and allow the pole to shift into the right half-plane (i.e., it begins oscillating)?
Typical operation in this regime is, a short pulse (or burst of pulses), repeating at a frequency much lower than Fsw, and may be repeating erratically (i.e., the error amp's noise is effectively magnified, and shifted up, into switching noise). The repeat rate being lower than Fsw is another way of saying the control loop has a RHP pole, that is about that far from zero.
Another aggravating factor is the minimum pulse width, which is usually limited by hysteresis in the PWM comparator, or driver, or of the current sense response, whether propagation delay of a peak current mode control, or bandwidth of an average current mode control. In the latter case, of course, you're more likely to get a burst of pulses.
In a lot of applications, the increased ripple doesn't matter -- it's still well within nominal range, because the error amp is otherwise doing its job just fine, and the switching noise is adequately filtered either way. Others, the consistent ripple from CCM may be easier to deal with (or filter) than the erratic DCM response, or the whine of the inductors or capacitors may be too objectionable.
CCM supplies still exhibit oscillation at very low output voltages -- minimum pulse width hits you regardless -- but the region out of the total SOA is very much smaller.

The downside is the constant "stirring" of reactive power, between input filter cap, and output filter choke. The reactive power, divided by the effective Q factor of this loop, equals switching loss.
Which by the way, is a handy way to think of switching supplies, that I don't think is mentioned very often; the switching ripple is reactive power, and the total Q factor of everything it flows through, gives losses. I used such an analysis here in regards to core Q:
https://www.seventransistorlabs.com/Articles/Core_Loss.html and plotted for a few materials here:
https://www.seventransistorlabs.com/Images/Powder_Core_Q.png The takeaway for lossy materials is, you avoid losses by using a small ripple fraction, which is to say using relatively few VARs for a given DC output. That's how you can manage to use a lossy powder core that dissipates 1W, for say 10 VAR switching (Q = 10), and 100W of real output.
Alternately if we have very low-loss reactive parts, we can stir some of the device reactives (most often Coss/Cjo) into the switching reactance, and voila, we have a resonant power supply that can run at much higher Fsw, and even achieve higher efficiency, than a conventional (square wave) design would (for the same Fsw).
Tim