Context. The annihilation of cosmic positrons with electrons in the interstellar medium results in the strongest persistent γ-ray line signal in the sky. For the past 50 yr, this 511 keV emission – predominantly from the galactic bulge region and from a low surface-brightness disk – has puzzled observers and theoreticians. A key issue for understanding positron astrophysics is found in cosmic-ray propagation, especially at low kinetic energies (≲10 MeV). Aims. We want to shed light on how positrons propagate and the resulting morphology of the annihilation emission. We approach this “positron puzzle” by inferring kinematic information of the 511 keV line in the inner radian of the Galaxy. This constrains propagation scenarios and positron source populations in the Milky Way. Methods. By dissecting the positron annihilation emission as measured with INTEGRAL/SPI, we derived spectra for individual and independent regions in the sky. The centroid energies of these spectra around the 511 keV line are converted into Doppler shifts, representing the line-of-sight velocity along different galactic longitudes. This results in a longitude-velocity diagram of positron annihilation. From high-resolution spectra, we also determined Doppler-broadening from γ-ray line shape parameters to study annihilation conditions as they vary with galactic longitude. Results. We found line-of-sight velocities in the 511 keV line that are consistent with zero, as well as with galactic rotation from CO measurements (2–3 km s⁻¹ deg⁻¹), and measurements of radioactive ²⁶Al (7.5–9.5 km s⁻¹ deg⁻¹). The velocity gradient in the inner ±30° is determined to be 4 ± 6 km s⁻¹ deg⁻¹. The width of the 511 keV line is constant as a function of longitude at 2.43 ± 0.14 keV, with possibly different values towards the disk. The positronium fraction is found to be 1.0 along the galactic plane. Conclusions. The weak signals in the disk leave the question open of whether positron annihilation is associated with the high velocities seen in ²⁶Al or rather with ordinarily rotating components of the Milky Way’s interstellar medium. We confirm previous results that positrons are slowed down to the 10 eV energy scale before annihilation and constrain bulk Doppler-broadening contributions to ≲1.25 keV in the inner radian. Consequently, the true annihilation conditions remain unclear.