Or can they just be used as an interpolation points and use some other "transported property" which are just evolved and propagated from boundary conditions like for eg. heat conduction through a solid metal bar. It seems to be possible, but isn't clearly elaborated. The force computation step is always used for fluid flow to move particles forward according to Newton's laws, but what if heat is transported purely by conduction in a solid? Here, nothing moves but still, heat flows and destroys temperature gradients. What equations are to be considered here (apart from the heat diffusion equation) to adapt this problem for this kind of discretization?
Particles in SPH simulations mimic the motion of material points by simply updating their positions due to their velocities. In situations like heat conduction you don't need to move them, in fact, there is no advection at all.
SPH is a collocation scheme like FDM (finite difference method) but without the rigid interpretation of nodes over a grid. The SPH meshless interpolant allows the evaluation of quantities regardless of the particle distribution (however, the interpolation error depends on the distribution). That is why SPH is powerful in fluid mechanics, where the distribution of the particles is changing in every instant.
To answer your question: You are right, you don't need to move the particles, but in such a case FDM is probably a better choice.
For more information about the solution of the heat conduction equation with SPH read this paper: J.H. Jeong, M.S. Jhon, J.S. Halow, J. van Osdol, Smoothed particle hydrodynamics: Applications to heat conduction, In Computer Physics Communications, Volume 153, Issue 1, 2003, Pages 71-84.