Context. Dwarf galaxies play an important role when studying the effects of the environment on galaxy formation and evolution. The Fornax cluster, having a dense core and strong tidal fields, offers an ideal laboratory for investigating the influence of the cluster environment on the morphology of dwarf galaxies. Aims. We explore the relationship between the morphology of galaxies, in particular the asymmetries, and their distances to the cluster centre to study the effect of tidal forces and other environmental processes. We did this by investigating the detailed morphologies of a complete magnitude-limited sample of 556 galaxies within the Fornax cluster, spanning a radius range up to 1.75 Mpc from its central to the outer regions. Methods. For galaxies in the Fornax Deep Survey, we quantified the morphologies of dwarf galaxies using the non-parametric quantities asymmetry (A) and smoothness (S), as part of the CAS system. Unlike previous work, we used isophotal CAS parameters, which are sensitive to the outer parts of galaxies. We constructed A - r (asymmetry vs. distance to cluster centre) and S - r (smoothness vs. distance to the cluster centre) diagrams to investigate the relationship between morphology and distance. Additionally, we examined the effects of asymmetry on magnitude and colour. Furthermore, to better understand the assembly history of the galaxy cluster, we performed a phase-space analysis for Fornax dwarf galaxies, using spectroscopic redshifts and the projected distance from the cluster centre. Results. We find that dwarf galaxies in the outer regions of the Fornax cluster have higher values of asymmetry compared to other dwarfs in the cluster, indicating a greater degree of morphological disturbances within dwarf galaxies in these regions. We also find that galaxies in the very inner regions are more asymmetric than those farther out. The A-magnitude relation reveals a trend where asymmetry increases as galaxies become fainter, and the A-colour relation shows that galaxies with bluer colours tend to exhibit stronger asymmetry. We do not find any correlations with smoothness, except that smoothness strongly decreases with stellar mass. We propose that the higher asymmetry of dwarfs in the outer regions is most likely caused by ram pressure stripping. As galaxies fall into the cluster, gas is expelled by intracluster winds, causing 'jellyfish-like' tails and leading to star formation not only in the central regions but also along the tails; this causes the asymmetric features. These asymmetries persist until the galaxies evolve into completely quiescent and elliptical systems. The observed dwarfs likely represent a transitional phase, during which they are nearing quiescence but still retain residual asymmetry from earlier interactions. In the very inner parts, the asymmetries most likely are caused by tidal effects. In addition, our phase-space diagram suggests that galaxies near pericentre in the Fornax cluster exhibit significantly higher asymmetry, indicating that morphological disturbances occurred during their first pericentric passage.