As a cornerstone of cloud computing and artificial intelligence, the Linux operating system has given rise to many popular distributions (distros), such as Fedora and Debian. A distro includes a rich collection of software packages derived from third-party open-source projects (upstream). Due to the nature of reusing upstream, some packages originate from the same upstream projects, noted as homologous packages. Maintaining packages is a core but tedious task for distros, which relies on continuously developing code patches. The maintenance of homologous packages could benefit from cross-distro collaboration. However, their shared origin may also lead to redundant efforts and delayed reuse of valuable maintenance work performed by other distros. To measure potential inefficiencies and explore possible solutions, this paper conducts an empirical study of patch sharing among homologous packages across four representative distros spanning three distinct families: Debian, Ubuntu, Fedora, and openEuler. We find that homologous packages are prevalent across distros, even across different distro families. Specifically, there are 13,873 homologous packages shared between Debian and Fedora, accounting for 40.3% and 56.1% of all packages in Debian and Fedora, respectively. For homologous packages with the same upstream version, we found that 19.5% of the patches are shared between Fedora and Debian, and 46.7% between Fedora and openEuler. However, these shared patches often exhibit remarkable propagation latency, with a median lag of 530.0 days between Fedora and Debian and 169.0 days between Fedora and openEuler. These results highlight the importance and necessity of cross-distro-family patch recommendations. We identified 17 shared Fedora-exclusive patches and contributed them to openEuler, among which 15 patches were successfully merged, demonstrating the potential of cross-distro-family patch recommendation and inspiring further research in this area.