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The physical mechanisms behind the fragmentation of high-mass dense clumps into compact star-forming cores and the properties of these cores are fundamental topics that are heavily investigated 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 (~2–8 kpc), masses (~10²–10⁴M_⊙), surface densities (0.1–10 g cm⁻²), and evolutionary stages (luminosity over mass ratio indicator of ~0.05 <L/M <450L_⊙/M_⊙). Here, we present the catalog of compact sources obtained with theCuTExalgorithm 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 ~1400 au (down to ~800 au). We characterize and discuss the revealed fragmentation properties and the photometric and 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 ~800–3000 au (median of 1700 au). We assigned core temperatures based on theL/Mof the hosting clump, and obtained core masses from 0.002 to 345M_⊙(complete above 0.23 M_⊙), exhibiting a good correlation with the core radii (M ∝ R^2.6). We evaluated the variation in the core mass function (CMF) with evolution as traced by the clumpL/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, while a population of possibly newly formed lower-mass cores is present throughout. The number of cores increases with the core masses, at least until the most massive core reaches ~10M_⊙. More generally, 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.