Compressive sensing (CS), acquiring and reconstructing signals below the Nyquist rate, has great potential in image and video acquisition to exploit data redundancy and greatly reduce the amount of sampled data. To further reduce the sampled data while keeping the video quality, this paper explores the temporal redundancy in video CS and proposes a block based adaptive compressive sensing framework with a sampling rate (SR) control strategy. To avoid redundant compression of non-moving regions, we first incorporate moving block detection between consecutive frames, and only transmit the measurements of moving blocks. The non-moving regions are reconstructed from the previous frame. In addition, we propose a block storage system and a dynamic threshold to achieve adaptive SR allocation to each frame based on the area of moving regions and target SR for controlling the average SR within the target SR. Finally, to reduce blocking artifacts and improve reconstruction quality, we adopt a cooperative reconstruction of the moving and non-moving blocks by referring to the measurements of the non-moving blocks from the previous frame. Extensive experiments have demonstrated that this work is able to control SR and obtain better performance than existing works.