Understanding vanishing transmission in Fano resonances in quantum systems and metamaterials and perfect and ultralow transmission in disordered media, has advanced the understanding and applications of wave interactions. Here we use analytic theory and numerical simulations to understand and control the transmission and transmission time in complex systems by deforming a medium and by adjusting the level of gain or loss. Unlike the zeros of the scattering matrix, the position and motion of the zeros of the determinant of the transmission matrix in the complex plane of frequency and field decay rate have robust topological properties. In systems without loss or gain, the transmission zeros appear either singly on the real axis or as conjugate pairs in the complex plane. As the structure is modified, two single zeros and a complex conjugate pair of zeros may interconvert when they meet at a square root singularity in the rate of change of the distance between the transmission zeros in the complex plane with sample deformation. The transmission time is the spectral derivative of the argument of the determinant of the transmission matrix. It is a sum over Lorentzian functions associated with the resonances of the medium, which is the density of states, and with the zeros of the transmission matrix. Transmission vanishes, and the transmission time diverges as zeros are brought near the real axis. Monitoring the transmission and transmission time when two zeros are close may open up new possibilities for ultrasensitive detection.