We consider the problem of recovering a sequence of vectors, (xk)k=0K, for which the increments xk−xk−1 are Sk-sparse (with Sk typically smaller than S1), based on linear measurements (yk=Akxk+ek)k=1K, where Ak and ek denote the measurement matrix and noise, respectively. Assuming each Ak obeys the restricted isometry property (RIP) of a certain order - depending only on Sk - we show that in the absence of noise a convex program, which minimizes the weighted sum of the ℓ1-norm of successive differences subject to the linear measurement constraints, recovers the sequence (xk)k=1K \emph{exactly}. This is an interesting result because this convex program is equivalent to a standard compressive sensing problem with a highly-structured aggregate measurement matrix which does not satisfy the RIP requirements in the standard sense, and yet we can achieve exact recovery. In the presence of bounded noise, we propose a quadratically-constrained convex program for recovery and derive bounds on the reconstruction error of the sequence. We supplement our theoretical analysis with simulations and an application to real video data. These further support the validity of the proposed approach for acquisition and recovery of signals with time-varying sparsity.