How fast can fire spread? You can build a home-made flamethrower by using a spray and a lighter:

(from this YouTube video)
But this can be dangerous if the fire gets in the can and the can explodes.
So I guess the speed at which the particles come out of the box has to be high enough to prevent this.

NOTE: I don't want to do this at home, because I think it's too dangerous. This is just a thought experiment!

I guess the question "how fast can fire spread?" highly depends on the details (like: What kind of exothermic reaction is going on; how big is the pressure at which the spray exits the can; temperature around the can). I would like one theoretical answer (choose everything to make it as fast as possible / slow as possible) and a practical answer (make a reasonable assumption of what could go on with the lighter / spray stuff).
 A: First, it's very unlikely that the fire will enter the can since there is no oxygen inside the can.
Second, the fire probably won't reach the nozzle since the fuel needs to vaporize before it can burn. It's a liquid inside the can. Although you could singe your fingers if you use the wrong fuel, or you have your finger over the front of the nozzle.
Once you have a flame, the lighter is no longer needed. Most demonstrations that I have seen use short bursts of fuel, probably to keep the flame manageable.
A: These notes supported by this thesis say the flame propagation in a hydrocarbon/air premix is $0.4-0.6$ m/sec.  Since yours is not premixed, it will be slower.  The aerosol will slow down quickly as it exits the can due to air resistance.
A: Using a commercial aerosol can to spray fuel is actually fairly safe, for two reasons, both of them mentioned by LDC3. At the nozzle exit, the fuel is liquid, so flame cannot propagate into the can. If, somehow, it were possible, the propellant used in aerosols is not flammable, so no combustion can occur inside the can. What can happen is that, with repeated sprays, a certain amount of fuel can build up on the can around the nozzle, and this can be ignited.
What is dangerous, at least in principle, is spraying fuel from a container that is not pressurized. The classic example is a can of lighter fluid or BBQ starter. In this case the fuel is expelled by squeezing the can, and when you release pressure the can sucks air into the can to replace the volume of fluid expelled. In principle, if the can is nearly empty the interior is full of fuel vapor/air mix, and burning fuel near the nozzle might get sucked back into the can if your timing were exactly right (or wrong, depending on your point of view), causing a fuel/air explosion in the can.
This is, as I say, in principle. I'm not certain it has actually happened. I'm dubious that combustion would actually pass the tiny nozzle opening. Miner's lamps in 19th century coal mines were protected by fine mesh, and the result was that methane concentrations (miner's damp) could not be ignited by the lamp flame, so I assume the same consideration applies here.
