There are really two sources of information you are talking about:
1) Why movers move is from the CPS-ASEC and you can readily estimate it at the state level using the IPUMS-CPS online analysis tool. Technically you can get MSA data, but the underlying sample will be too small to support analysis for most MSAs.
2) Demographic characteristics of move to vs. move from are available down to the MSA or county level from the ACS. While the summary table you are linking is just the "move to," it's made by compiling a variety of underlying tables on geographic mobility in the past year by select characteristics. I think all of the mobility tables begin with B074. Tables that are for "current residence" represent move to; tables for "residence 1 year ago" represent move from. Together, you can look at in-migrants, out-migrants, and net-migrants by select demographic characteristics.