How is propagation of action potentials terminated once the "message" has been sent/received?
I'm not a Neurologist or Neuroscientist, so I can only share what I know about the subject!

TL:DR


Cease sensation
Neuronal recovery period prevents further signaling
Neurotransmitter inhibition prevents ligands from binding to receptors (usually Feedback mechanisms)
Neuronal hyperpolarization prevents signaling
Creation of signal (can be perceived as cessation of signal)


It's important to note, first and foremost, that a cessation of stimulation will result in a cessation of the propagation. This is one of the simplest ways that neurons can stop firing - you're no longer touching, tasting, hearing, seeing, smelling, or thinking of what you were previously. 

Because of the way Neurons are constructed, and as far as I'm aware, signal propagation will persist as far down the line as possible. I don't believe there's a discrete point where an Axon will simply not send a signal once a dendrite on the same neuron has received the order.

What can happen is that a competing signal has exhausted the propagation potential for the next neuron in the chain (the neuron is still recovering), or a competing signal has released inhibitors that prevents the neurotransmitters from another Axon from affecting the target neuron. Competing signals is particularly common in the neurons of your eye, since the Amacrine cells fire depending on the competitive input of the Cones if I'm remembering everything correctly.



Then there are also substances like Benzodiazepines which block propagation by opening Cl- channels and hyperpolarizing a neuron's membrane, which prevents it from firing. Only a subset of GABA receptors have sites for benzodiazepines, but it's one more way that signals will cease.

Another way to cease a signal, or at least the perception of a signal, is to actually induce propagation. The Rods in your eyes only fire below a certain lumination threshold. What you perceive as light is them not firing, and darkness is what happens when Rods fire at full strength. 

I'm not the last word on neurobiology, but I hope I've given you a good start!