Not yet reviewed by Pith; the record is open.

The mechanisms behind the fragmentation of high-mass dense clumps into compact star-forming cores are fundamental topics in current astrophysical research. The ALMAGAL survey provides the opportunity to study this process at an unprecedented level of detail and statistical significance, featuring high-angular resolution $1.38$ mm ALMA observations of $1013$ massive dense clumps at various Galactic locations. These clumps cover a wide range of distances, masses, surface densities, and evolutionary stages. Here, we present the catalog of compact sources obtained with the CuTEx algorithm from continuum images of the full ALMAGAL clump sample combining ACA-$7$m and $12$m ALMA arrays, reaching a uniform high median spatial resolution of $\sim1400$ au. We discuss the fragmentation properties and the estimated physical parameters of the core population. The ALMAGAL compact source catalog includes $6348$ cores detected in $844$ clumps ($83\%$ of the total), with a number of cores per clump between $1$ and $49$ (median of $5$). The estimated core diameters are mostly within $\sim800-3000$ au (median of $1700$ au). We obtained core masses from $0.002$ to $345\,\mathrm{M_{\odot}}$. We evaluated the variation in the core mass function (CMF) with evolution as traced by the clump $L/M$, finding a clear, robust shift and change in slope among CMFs within subsamples at different stages. This finding suggests that the CMF shape is not constant throughout the star formation process, but rather it builds (and flattens) with evolution, with higher core masses reached at later stages. We found that all cores within a clump grow in mass on average with evolution, and the number of cores increases with the core masses. Our results favor a clump-fed scenario for high-mass star formation, in which cores form as low-mass seeds, and then gain mass while further fragmentation occurs in the clump.