A small group of the newly discovered superluminous supernovae show broad and slow-evolving light curves. Here we present extensive observational data for the slow-evolving superluminous supernova LSQ14an, which brings this group of transients to four in total in the low-redshift Universe ($z$ < 0.2; SN 2007bi, PTF12dam, SN 2015bn). We particularly focus on the optical and near-infrared evolution during the period from 50 d up to 400 d from peak, showing that they are all fairly similar in their light curve and spectral evolution. LSQ14an shows broad, blueshifted [O $III$] λλ4959, 5007 lines, as well as a blueshifted [O $II$] λλ7320, 7330 and [Ca $II$] λλ7291, 7323. Furthermore, the sample of these four objects shows common features. Semi-forbidden and forbidden emission lines appear surprisingly early at 50–70 d and remain visible with almost no variation up to 400 d. The spectra remain blue out to 400 d. There are small, but discernible light-curve fluctuations in all of them. The light curve of each shows a faster decline than $\mathrm{<mrow\; data-mjx-texclass="ORD">56</mrow>}$Co after 150 d and it further steepens after 300 d. We also expand our analysis presenting X-ray limits for LSQ14an and SN 2015bn and discuss their diagnostic power. These features are quite distinct from the faster evolving superluminous supernovae and are not easily explained in terms of only a variation in ejecta mass. While a central engine is still the most likely luminosity source, it appears that the ejecta structure is complex, with multiple emitting zones and $at\; least$ some interaction between the expanding ejecta and surrounding material.