In this thesis we investigate discrete space Markov processes with multiple layers and how these can be applied to physical systems such as RNA transcription. We show that the Markov processes of single particles satisfy an invariance principle (i.e., the limiting behaviour is Brownian with a certain drift) for both homogeneous and random environments. Additionally, we perform an in-depth numerical analysis of multiple particle systems in the context of RNA transcription and reproduce some known phenomena such as RNA polymerase clustering and cooperation.