According to a generally accepted paradigm, small intrinsic sizes of Compact Steep Spectrum (CSS) radio sources are a direct consequence of their youth, but in later stages of their evolution they are believed to become large-scale sources. However, this notion was established mainly for strong CSS sources. In this series of papers we test this paradigm on 60 weaker objects selected from the VLA FIRST survey. They have 5-GHz flux densities in the range 150 < S_{5GHz} < 550 mJy and steep spectra in the range 0.365 le u le 5 GHz. The present paper is focused on sources that fulfill the above criteria and have angular sizes in the range 0.2arcsec - 1arcsec. Observations of 19 such sources were obtained using MERLIN in snapshot mode at 5 GHz. They are presented along with 1.7-GHz VLBA and 5-GHz EVN follow-up snapshot observations made for the majority of them. For one of the sources in this subsample, 1123+340, a full-track 5-GHz EVN observation was also carried out. This study provides an important element to the standard theory of CSS sources, namely that in a number of them the activity of their host galaxies probably switched off quite recently and their further growth has been stopped because of that. In the case of 1123+340, the relic of a compact dead source is particularly well preserved by the presence of intracluster medium of the putative cluster of galaxies surrounding it. The observed overabundance of compact sources can readily be explained in the framework of the scenario of premature cessation of the activity of the host galaxy nucleus. It could also explain the relatively low radio flux densities of many such sources and, in a few cases, their peculiar, asymmetric morphologies. We propose a new interpretation of such asymmetries based on the light-travel time argument.
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