We present the results of a Hubble Space Telescope (HST) study of dwarf galaxies in the outer regions of the nearby rich Perseus cluster, down to MV=-12, and compare these with the dwarf population in the cluster core from our previous HST imaging. In this paper, we examine how properties such as the colour-magnitude relation, structure and morphology are affected by environment for the lowest mass galaxies. Dwarf galaxies are excellent tracers of the effects of environment due to their low masses, allowing us to derive their environmentally based evolution, which is more subtle in more massive galaxies. We identify 11 dwarf elliptical (dE) and dwarf spheroidal (dSph) galaxies in the outer regions of Perseus, all of which are previously unstudied. We measure the (V-I)0 colours of our newly discovered dEs, and find that these dwarfs lie on the same red sequence as those in the cluster core. The morphologies of these dwarfs are examined by quantifying their light distributions using concentration, asymmetry and clumpiness (CAS) parameters, and we find that dEs in the cluster outskirts are on average more disturbed than those in the core, with (A)= 0.13 A? 0.09 and (S)= 0.18 A? 0.08, compared to (A)= 0.02 A? 0.04, (S)= 0.01 A? 0.07 for those in the core. Based on these results, we infer that these objects are transition dwarfs , likely in the process of transforming from late-type to early-type galaxies as they infall into the cluster, with their colours transforming before their structures. When we compare the number counts for both the core and outer regions of the cluster, we find that below MV=-12, the counts in the outer regions of the cluster exceed those in the core. This is evidence that in the very dense region of the cluster, dwarfs are unable to survive unless they are sufficiently massive to prevent their disruption by the cluster potential and interactions with other galaxies.