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We present a method of kinematic analysis of proper motions by vector spherical functions, and the results of its application to astrometric data. The sets of vector spherical functions which are orthonormal on a full sphere as well as on a latitude zone are constructed. Decomposition of the proper motions into a set of such functions allows model-independent study of stellar kinematics. If needed, the parameters of the standard (say, Ogorodnikov-Milne) model may be derived from the coefficients of the decomposition. In contrast to the commonly used least squares estimation of the model's parameters, vector spherical functions identify all systematic components of the velocity field (no matter, whether they are incorporated into the model or not) and give us a possibility to test whether the data are compatible with the model. In this paper, we apply this technique for the first time to the proper motions from the UCAC4 catalog for stars in the 11 to 16 magnitude range. We derive all-sky solutions and the solutions based on stars in the northern and southern Galactic hemispheres. The all-sky solution provide evidence for noticeable magnitude-dependent trends in the coordinates of the solar motion apex, Oort constants, angular speed of the local Galactic rotation, and the slope of the local rotation velocity curve as we go from bright to faint stars. Furthermore, our all-sky vector spherical function analysis identified strong and reliable extramodel harmonics, whereas the solutions for the northern and southern hemisphere indicate sign reversals for some of the Ogorodnikov-Milne parameters. We show that both effects appear simultaneously and can be explained by the slowdown of Galactic rotation with increasing distance from the main Galactic plane. We estimate the absolute value of the vertical gradient of the Galactic rotational velocity to be ∼ 40 km s −1 kpc −1 .