A very significant application of such a setup in the context of communications engineering is the modelling of multiple-input-multiple-output (MIMO) communications channels.
These channels are typically modeled by complex $n \times m$ matrices where $n$ is the number of receive antennas and $m$ is the number transmit antennas. The $(i,j)$ entry in the matrix describes the channel between the $i$th transmit antenna and the $j$th receive antenna. In most applications $n$ and $m$ are reasonably small, less than 16. Also, in most real world applications the channel (and hence the matrix) changes over time. This gives you your time series of matrices. In some situations the matrix will even be sparse because some transmit antennas might not see some receive antennas.
There is a seriously huge amount of literature on the MIMO channel and a large amount of it deals with the static case, i.e. for the sake of simplicity it is assumed that the channel doesn't change with time. However there are also many papers that deal with the time varying case. For example:
I unfortunately am not an expert in MIMO, but I do know some people who are and could ask them for more details if you were interested.