Question: Ok, so I'm willing to bet at least one person has looked at the title of this question and thought to themselves "What?", so I'll try and explain this as thoroughly as I can.

Most incarnations of Pyrokinesis I've seen in fiction often deal with either telekinetically starting fires by exciting the flammable properties within the objects around them, or just straight up creating and releasing fire from within a person's body (the science behind this particular incarnation of the ability often varies from each piece of work). Some pyrokinetic people in fiction however, are capable of moving and shaping fires that are around them, sorta like how Aerokinesis lets you move air/wind and apply a shape to it.

Since fire is nothing but the result of a flammable material/substance getting exposed to oxygen and heat, this got me thinking. What are pyrokinetic people actually moving when they use their powers? Calling it fire seems inaccurate seeing as how fire itself lacks weight, cannot exist without certain requirements, and is treated less like a 'thing' but more so a chemical reaction. My understanding on science is how you say "Painfully mediocre", so I wonder if any of you guys and girls could help me out?

Answer: 
**Moving a match**

How do you normally move fire? You might shape the fire with funnels or wind, but the starting point is always the same. The flammable material. If you want to move the fire of a match, you move the match. Moving the match moves the flammable material, which in turn moves the origin point of the flame.

With telekinesis you want to move the flammable material as well, be it oil fumes or burning wood. Shaping of flame can be done afterwards with wind and funnels, which is basically pressure the flame cannot overcome. Telekinesis can so this as well while mindful of the limitations of the flame, like reach, starting point and duration.
What is fire? Fire happens normally when objects oxidize quickly, but in the case of pyrokinesis / telekinesis of fire, in my mind, there is an implication of the ability of the caster to facilitate the transfer of energy between molecules. Therefore, when someone is "telekinetically moving/shaping fire," they are transferring energy from the particles in one region of space to the particles in another region of space. So similar to what @DWKraus says in their comment, telekinesis of fire should allow the user to move energy to cause molecular heating.

However, I think generalizing it in this way has its benefits, and can potentially take the idea of moving/shaping fire to the next level. What happens, for example, when energy is concentrated in a certain location? Since kinetic energy actually has mass (and since mass and energy are equivalent by E=mc^2), moving/shaping fire could even be the basis for abilities that control mass or gravity, or even slow and alter the movement of time, given the connection that gravity has to time. Makes the ability a lot more interesting to think of it this way, but that's just my 2 cents