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Photometrically distinct nuclear star clusters (NSCs) are common in late-type-disk and spheroidal galaxies. The formation of NSCs is inevitable in the context of normal star formation in which a majority of stars form in clusters. A young, mass-losing cluster embedded in an isolated star-forming galaxy remains gravitationally bound over a period determined by its initial mass and the galactic tidal field. The cluster migrates radially toward the center of the galaxy and becomes integrated in the NSC if it reaches the center. The rate at which the NSC grows by accreting young clusters can be estimated from empirical cluster formation rates and dissolution times. We model cluster migration and dissolution and find that the NSCs in late-type disks and in spheroidals could have assembled from migrating clusters. The resulting stellar nucleus contains a small fraction of the stellar mass of the galaxy; this fraction is sensitive to the high-mass truncation of the initial cluster mass function (ICMF). The resulting NSC masses are consistent with the observed values, but generically, the final NSCs are surrounded by a spatially more extended excess over the inward-extrapolated exponential (or Sersic) law of the outer galaxy. We suggest that the excess can be related to the pseudobulge phenomenon in disks, though not all of the pseudobulge mass assembles this way. Comparison with observed NSC masses can be used to constrain the truncation mass scale of the ICMF and the fraction of clusters suffering prompt dissolution. We infer truncation mass scales of ~ 10^5 M_sun) without (with 90%) prompt dissolution.