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Context. Interstellar dust is a key element in our understanding of the interstellar medium and star formation. The manner in which dust populations evolve with the excitation and the physical conditions is a first step in comprehending the evolution of interstellar dust. Aims. Within the framework of the Evolution of interstellar dust Herschel key programme, we have acquired PACS and SPIRE spectrophotometric observations of various photodissociation regions, to characterise this evolution. The aim of this paper is to trace the evolution of dust grains in the Orion Bar photodissociation region. Methods. We used Herschel/PACS (70 and 160 μm) and SPIRE (250, 350 and 500 μm) together with Spitzer/IRAC observations to map the spatial distribution of the dust populations across the Bar. Brightness profiles were modelled using the DustEM model coupled with a radiative transfer code. Results. Thanks to Herschel, we are able to probe in great detail the dust emission of the densest parts of the Orion Bar with a resolution from 5.6'' to 35.1'' . These new observations allow us to infer the temperature of the biggest grains at different positions in the Bar, which reveals a gradient from ∼70 K to 35 K coupled with an increase of the spectral emissivity index from the ionization front to the densest regions. Combining Spitzer/IRAC observations, which are sensitive to the dust emission from the surface, with Herschel maps, we were able to measure the Orion Bar emission from 3.6 to 500 μm. We find a stratification in the different dust components that can be quantitatively reproduced by a simple radiative transfer model without dust evolution (diffuse interstellar medium abundances and optical properties). However, including dust evolution is needed to explain the brightness in each band. Polycyclic aromatic hydrocarbon (PAH) abundance variations, or a combination of PAH abundance variations with an enhancement of the biggest grain emissivity caused by coagulation give good results. Another hypothesis is to consider a length of the Bar along the line of sight different at the ionization front than in the densest parts. [ABSTRACT FROM AUTHOR]