A popular definition from Wikipedia for karyorrhexis is that it is "the destructive fragmentation of the nucleus of a dying cell whereby its chromatin is distributed irregularly throughout the cytoplasm. It is usually preceded by pyknosis and can occur as a result of either programmed cell death, senescence, or necrosis". Wikipedia
No doubt this is the appropriate definition for most of the more than 1000 entries in PubMed elicited using karyorrhexis as the search term. This website confirms that the definition is definitely not true is all instances of the process.
For differentiation of epithelial cells into domes, the process of karyorrhexis is definitely observed, but the outcome of the process requires changes in the definition as follows.
In differentiation, karyorrhexis is the dramatic fragmentation of an aggregated polyploidal nuclear structure involved in differentiation of domes/glands whereby chromatin fragmented into granules/filaments is arrayed throughout the cytoplasm for a short period of time (from the rarity of images such as that shown in fig. 7, probably measured in minutes).
Fig. 7 Polyploid nuclei are fragmenting into granules/filaments and the granules are moving through the cytoplasm in relatively straight lines, pulled, it appears, by centriole like structures. Thus begins the formation of a chromatin array that open up chromatin to changes brought about by enzymes or by association of DNA with new proteins or altered histones. The altered chromatin will be recycled into dome/gland nuclei capable of synthesizing and secreting proteins into a lumen in cells that are anchorage independent.
Fig. 8. Because of the limits of the light microscope, fixing and staining such a structure does not add too much to the information from the previous photomicrograph. No nucleus is visible. Chromatin and heterochromatin are staining at both polar ends of the structure. Faint images of stained strands appear to stretch between these two structures.
From electron microscopic studies done with normal human endometrium, structures characterized by chromatin fragments are observed in endometrium between days 12 and 18, at around the time of ovulation when the endometrium enters the secretory phase, suggesting that they signal the differentiation of proliferating epithelial cells into glands.
In a salient review of many more examples of DNA fragmentation not associated with apoptosis, Sjaste & Sjaste (2007) hypothesize that DNA fragmentation is an epigenetic tool for regulation of the differentiation process.
Sjakste N1, Sjakste T Possible involvement of DNA strand breaks in regulation of cell differentiation. Eur J Histochem. 2007 Apr-Jun;51(2):81-94.