Dwarf spheroidal galaxies (dSphs) have been extensively investigated in the Local Group, but their low luminosity and surface brightness make similar work in more distant galaxy groups challenging. Modern instrumentation unlocks the possibility of scrutinizing these faint systems in other environments, expanding the parameter space of group properties. We use MUSE spectroscopy to study the properties of 14 known or suspected dSph satellites of Cen A. Twelve targets are confirmed to be group members based on their radial velocities. Two targets are background galaxies at ∼50 Mpc: KK 198 is a face-on spiral galaxy, and dw1315−45 is an ultra-diffuse galaxy with an effective radius of ∼2300 pc. The 12 confirmed dSph members of the Cen A group have old and metal-poor stellar populations and follow the stellar metallicity-luminosity relation defined by the dwarf galaxies in the Local Group. In the three brightest dwarf galaxies (KK 197, KKs 55, and KKs 58), we identify globular clusters, as well as a planetary nebula in KK 197, although its association with this galaxy and/or the extended halo of Cen A is uncertain. Using four discrete tracers, we measure the velocity dispersion and dynamical mass of KK 197. This dSph appears dark matter dominated and lies on the radial acceleration relation of star-forming galaxies within the uncertainties. It also is consistent with predictions stemming from modified Newtonian dynamics. Surprisingly, in the dwarf KK 203 we find an extended Hα ring. Careful examination of Hubble Space Telescope photometry reveals a very low level of star formation at ages between 30 and 300 Myr. The Hα emission is most likely linked to a ∼40 Myr old supernova remnant, although other possibilities for its origin cannot be entirely ruled out. Key words: galaxies: dwarf / galaxies: kinematics and dynamics / galaxies: stellar content / Galaxy: abundances⋆ The fully reduced data cubes are only available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/cat/J/A+A/645/A92⋆⋆ Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO program 0101.A-0193(A) and 0101.A-0193(B).