Dynamic functional connectivity (DFC) analysis has been widely applied to functional magnetic resonance imaging (fMRI) data to reveal time-varying dynamic changes of brain states. The sliding window method is by far the most popular DFC analysis method due to its simplicity. However, the sliding window method comes with some assumptions, namely the typically approach uses a single window which captures dynamics only within a specific frequency range. In this study, we propose a generalized approach to dynamics via a multi-dimensional random convolution (RandCon) DFC method that is able to effectively capture time-varying DFC at arbitrary time scales by extracting different local features from fMRI time series using a number of multi-dimensional random convolution kernels without the need for learning kernel weights. Compared to a standard sliding window method, multiplication of temporal derivatives (MTD) and phase synchrony methods, RandCon with the smallest kernel size (3 time points) showed notable improvements in performance on simulated data, particularly in terms of DFC temporal and spatial estimation in very short window/kernel size under different noise levels. Results from real fMRI data indicated that RandCon was more sensitive to gender differences than competing methods. Furthermore, we show that the sliding window method can be considered a special case of the proposed multi-dimensional convolution framework. The proposed method is simple and efficient significantly broadens the scope of dynamic functional connectivity research and offer theoretical and practical potential.