  Observations of the off-limb corona with the SWAP wide-field telescope in the
174\,\AA\ passband onboard the PROBA2 mission provide an opportunity to study
post-eruptive processes in the dimming regions. We investigate morphology,
temporal evolution, and plasma properties for four `deep' off-disk coronal
dimmings associated with limb coronal mass ejections (CMEs) in 2010\,--\,2017.
Using the SWAP fixed-difference images, we revealed ray-like structures that
appeared in the dimming recovery phase stretching quasi-radially to distances
from 1.1 to 1.6\,$\mathrm{R}_{\odot}$ and existing from tens of minutes to
several hours. Similar rays were detected earlier at distances above
1.7\,$\mathrm{R}_{\odot}$ by the UVCS onboard the SOHO. These structures
apparently represent the coronal roots of the flux rope trunks observed by the
LASCO-C2 onboard SOHO. The EUVI data onboard the STEREO show the origins of
these structures on the disc as fan rays much brighter in the 171\,\AA\ band
than in 193\,\AA , which suggests their temperature being less than 2\,MK. The
differential emission measure (DEM) analysis based on the AIA multi-wavelength
images onboard the SDO showed that the emission measure (EM) in these rays
compared to the pre-eruption plasma state increased up to 45\,\% at
temperatures of 0.6\,--\,0.8\,MK, whereas EM of the ambient coronal plasma with
temperatures of 1.3\,--\,3.7\,MK dropped by 19\,--\,43\,\%. For the event on 18
August 2010, the PLASTIC instrument onboard STEREO detected signatures of the
cold streams in the CME tail as enriched with the ions
Fe$^{8+}$\,--\,Fe$^{10+}$, which may be associated with the post-eruptive rays
in the solar corona.
