That is simply convection.
The wick does suck molten wax, and it goes up by capillary to the middle of the flame, but that movement is way to slow to explain the fast particles in your video. Moreover they are moving in the opposite direction!
Convection happens because the wick is hot, and it makes the wax around also hot, so the wax expands a little and gets lighter. Since hot wax is lighter it will go up (because of buoyancy). And since it goes up around the wick, it will go to the sides when it reaches the surface, then cools in contact with air and goes down.
Note how the red arrows curve to the sides when they reach the surface. That is what you see in the moving particles.
In your video you can even see the black particles moving towards the wick when they go down, doing full circles, just as in the graphic.
Usually, in convection the heat comes from below, as in this graphic from Wikipedia. The effect is similar, but instead of having the heat source at the bottom, it would be just in between two red arrows.
Combustion is... complicated. Essentially what is going on in flame is that you have molecules of fuel and oxidizer that mix and start to bounce off each other. If the molecules are moving fast enough (meaning they have enough energy, which we measure as temperature), then when they collide with each other, they start to make the fuel and oxidizer fall apart into other molecules.
Depending on which molecules collide and the energies involved, when things start to fall apart they are moving to lower energy states and the energy that was stored in the chemical bonds gets released as heat (and radiation in the form of light, which may be invisible). If it is happening often enough, the heat raises the temperature (adds energy) of the molecules around it and the process starts to run away. This is how you get a stable flame.
So this means there's at least a fundamental limit to the thickness of a flame -- you couldn't have a flame at lengths smaller than the distance molecules travel before they collide. This distance is called the mean free path, but frankly it's not a useful limit because flames cannot exist on the scale of the mean free path for other reasons.
For a flame to exist and be stable (i.e. not just a spark or something that goes away quickly), the rate of heat release has to be in balance with the rate of heat losses. If heat release exceeds heat losses, the flame will get bigger. If heat release is less than heat losses, the flame will run out of energy.
All of this means it is difficult, if not impossible, to put a general limit on the smallest possible flame. It will depend on the fuel source and how much oxidizer is present (different fuel+oxidizer combinations need different energies to start releasing heat), what the flow around the flame is like (how fast heat is carried away), how much energy the mixture has (higher temperature means more collisions that can break things apart), and how far the molecules need to move before they collide (how dense the mixture is).
The only definitive thing we can say is that the flame needs to be thicker than the mean free path, but anything more precise would require getting specific about the setup.
For a candle flame, we're looking at what is called a diffusion flame. The fuel (wax) is on one side and it has to vaporize and diffuse/mix with the oxygen in the air before it can properly burn. This is pretty hard. An overview lecture on diffusion flames is available, but it's actually not that easy to define a thickness for diffusion flames.
Suffice to say that the flames can be arbitrarily small, at least for sizes greater than several times the mean free path, provided the heat release is in balance with the heat losses. To be more specific would require a lot more details of the setup.
Best Answer
The factors that most matter when you are near lava:
In essence, if we treat lava as a black body radiator with an emissivity of 0.8 (just to pick a "reasonable" value), we can compute the heat flow to an observer. This is essentially a fraction of the heat flow you would have if you were completely surrounded on all sides. This means that if you have a semi-infinite plane of lava, your height as an observer will matter a great deal - if you crouch down, the plane "looks smaller" and you will experience less heat flux. When you stand up, your head will get more heat than the rest of you.
Temperature: radiated power goes as the fourth power of temperature, so this is the most important number to estimate correctly. A 10 % change (say from 800 to 900 C) results in a 40% change in radiation. Google gives values from 800 (Mt St Helens) to 1100 (Hawaiian basalt) so there is a lot of variability here
Reflectivity: assume you wear white clothes (looks better in the movie) you might reflect 80% of the incident radiation
Air flow: if there is a bit of wind blowing to cool you down, that will help. Luckily, if you are on the edge of a lava field, the effect of the heat will be to draw cold air in and then lift it up - so you should have a cool breeze (I have never been near a lava field but I think that's a reasonable speculation)
Toxic fumes: if the above is true, the effect of toxic fumes will be mitigated by the built in "extractor fan" formed by the heat.
Calculating: assume a height $h$ at distance $d$ from a semi infinite plane at temperature $T$:
Heat flux per unit area of the lava (Stefan-Boltzmann law)
$$F = \epsilon \sigma T^4$$
Fraction of solid angle covered (I think this approximation is valid... there may be a factor 2 gone astray):
$$f = \frac{\tan^{-1}\frac{h}{d}}{2\pi}$$
Apparent heat flux at observer (taking into account reflectivity $r$ and emissivity $\epsilon$):
$$F_{obs}= \epsilon \sigma T^4 (1-r) \frac{\tan^{-1}\frac{h}{d}}{2\pi}$$
The intensity of the sun on earth's surface is about $1 kW/m^2$. Let's assume that you are OK when you are receiving five times that (just to get an order of magnitude). Then we need to solve for $h/d$ in the above (let's use hot lava - 1300 K):
$$\frac{h}{d} = tan{\frac{2\pi 5\cdot 10^3}{0.8 \cdot 5.6 \cdot 10^{-8} 1300^4 (0.2)}}$$
This results in an angle of about 50 degrees. That's interesting - it suggests that if you get close to the lava but crouch down, you should be OK. But if you stand up, the fact you are "looking at" so much lava burns you. Put differently - if you are 1.80 m ("six feet") tall, then you are OK when you are at least 2 m from the edge of the lava - for all the above assumptions. Note that reflectivity does play directly into this calculation - if you don't wear a reflective face mask, the heat of 25 suns will be bearing down on you, and that may be too much... In which case you need an angle around 10 degree - or stand about 10 m away.
Of course there are secondary effects of heat absorption etc - but this is actually quite an interesting result.