Anomalous cross-field electron transport in a specified magnetic field with weakly-destroyed flux surfaces is discussed. Following the approach developed previously in the present paper the regimes of effective transverse plasma transport are studied systematically, both in the collisional and collisionless limits. The present analysis incorporates the nonstationary of magnetic perturbations, which was not included in the previous works. Part I of this study deals with the quasi-linear approximation, which may be written as b0L0/ delta <<1, where b0= delta Bperpendicular to /B0 represents the relative magnitude of the transverse magnetic perturbation, while L0 and delta represent the longitudinal and transverse correlation lengths, respectively. It is found that some of the previously-described transport regimes cannot be considered as anomalous (in the sense that effective heat conduction chi eff>> chi perpendicular to ) for time-independent magnetic perturbations. However, these regimes can exist in the presence a finite-frequency magnetic flutter. A unified classification of quasi-linear regimes of anomalous transport is introduced in order to further extend the analysis to the strong magnetic turbulence limit b0L0/ delta >>1 (see Part II), which is considered as a companion paper. (Isichenko, ibid., vol.33, no.7, p.809-26 (1991)).