Which elementary particles are behind magnetic field, similar as photons behind radio waves? I see, there are photons behind radio waves. 
As Wave–particle duality said: the radio waves are waves and at the same time are fluxes of particles called Photons.
I'm wondering, what is behind magnetic fields? 
Means, magnetic fields - are waves too. Shall this waves consists of photons too? Then how does photons moves by such trajectories?? 

 A: First. Eletric and magnetic field is dependent of the inertia frame. Only when we see this two fields together, that we have an physical object living in space-time, being independent of the inertial frame. Actually, when we do this we learn what is the truly symmetries of our space-time. Einstein is the one that discover that ;).
Radio waves are compound by electric and magnetic field. One create the another through space and time. See Maxwell's equations that describes this process.
Let's go to you question:
Behind the electromagnet (EM) field we have photons. But this is in some way a quantum relationship. We can say that photons and EM field is two faces of the same thing, as a generalization of wave-particle duality. In some cases we may have a bunch of photons propagating through space building an EM wave. In other cases we may have a state builded by a lot of superpositions of photons created from the vacuum propagating through space and annihilated to the vacuum, that constructively build an stationary EM field state. Actually the vacuum itself is build in that way. This process is more clear when you understand Feynman diagrams, that is a perturbative application of Path integral formulation of Quantum Mechanics.
This is quantum mechanics, so don't think that this particles are actually there. Only if you put an delicate detector you would measure they. But in quantum mechanics before the measure we have only potentialities of possible outcomes and they probabilities. And after the measurement you destroy the field state or vacuum state, in case of local measurements. 
A: A particle must contain some amount of energy. Otherwise it wouldn't exist. There is no energy transfer with a steady-state magnetic (or electric) field, so there are no particles involved in transferring energy. Only when a magnetic field changes (the producer of that field moves) can there be energy transferred. In that case, the particle transferring the energy is the photon.
A: 
magnetic fields are waves too - shall this consist of photons too?

The electric and magnetic fields were quite early on unified into the electromagnetic field.
Photons are then the quanta of this EM field; so this covers both cases.
