The flow of viscous incompressible fluid in a circular cone induced by a non-zero velocity prescribed at the boundary within a ring $0\,{<}\,a_1\,{<}\,r\,{<}\,a_2\,{<}\,\infty$, where $r$ is the distance from the vertex, is considered in the limits of the Stokes approximation. In the spherical coordinate system $(r,\theta,\phi)$ with the origin at the vertex and the axis $\theta\,{=}\,0$ coincident with the axis of the cone the velocity and pressure fields are represented in the form of a Fourier series on the trigonometric system $\cos m \phi$. The solution is constructed for each term by use of the Mellin transform. The contribution of each term of the Fourier expansion to the local velocity field near the vertex is studied. The kinematics of the local flows is illustrated by two examples. The flows are induced by the motion of two and three equally spaced segments, respectively.