We investigate the origin of stars in the innermost 500 pc of galaxies spanning stellar masses of $5\times 10^{8-12}\, \mathrm{M}_{\odot }$ at z = 0 using the cosmological magnetohydrodynamical TNG50 simulation. Three different origins of stars comprise galactic centres: (1) in situ (born in the centre), (2) migrated (born elsewhere in the galaxy and ultimately moved to the centre), (3) ex situ (accreted from other galaxies). In situ and migrated stars dominate the central stellar mass budget on average with 73 and 23 per cent, respectively. The ex situ fraction rises above 1 per cent for galaxies $\gtrsim 10^{11}\, \mathrm{M}_{\odot }$. Yet, only 9 per cent of all galaxies exhibit no ex situ stars in their centres and the scatter of ex situ mass is significant ($4\!-\!6\, \mathrm{dex}$). Migrated stars predominantly originate closely from the centre ($1\!-\!2\, \mathrm{kpc}$), but if they travelled together in clumps distances reach $\sim \! 10\, \mathrm{kpc}$. Central and satellite galaxies possess similar amounts and origins of central stars. Star-forming galaxies ($\gtrsim 10^{10}\, \mathrm{M}_{\odot }$) have on average more ex situ mass in their centres than quenched ones. We predict readily observable stellar population and dynamical properties: (1) migrated stars are distinctly young ($\sim \! 2\, \mathrm{Gyr}$) and rotationally supported, especially for Milky Way-mass galaxies, (2) in situ stars are most metal-rich and older than migrated stars, (3) ex situ stars are on random motion dominated orbits and typically the oldest, most metal-poor and α-enhanced population. We demonstrate that the interaction history with other galaxies leads to diverse pathways of building up galaxy centres in a Lambda cold dak matter universe. Our work highlights the necessity for cosmological context in formation scenarios of central galactic components and the potential to use galaxy centres as tracers of overall galaxy assembly.