The recently proposed extreme-ultraviolet beams with time-varying orbital angular momentum (OAM) realized by high-harmonic generation provide extraordinary tools for quantum excitation control and particle manipulation. However, such an approach is not easily scalable to other frequency regimes. We design a space-time-coding digital metasurface operating in the microwave regime to experimentally generate time-varying OAM beams. Due to the flexible programmability of the metasurface, a higher-order twist in the envelope wavefront structure of time-varying OAM beams can be further designed as an additional degree of freedom. The time-varying OAM field patterns are dynamically mapped by developing a two-probe measurement technique. Our approach in combining the programmability of space-time-coding digital metasurfaces and the two-probe measurement technique provides a versatile platform for generating and observing time-varying OAM and other spatiotemporal excitations in general. The proposed time-varying OAM beams have application potentials in particle manipulation, time-division multiplexing, and information encryption.