Have you tried making univariate control charts of all the variables that went into the multivariate control chart and look for trends, shifts etc. in each of those? Then you should be able to apply typical root cause analysis methods.
I would look at the univariate "control" (i.e., run) charts for the individual variables but remember that you might not find any single factor out of control as the out of control may be due to the multivariate structure in the data. You may have to "dig" a bit further. I would recommend "Multivariate Statistical Process Control with Industrial Applications" by Mason and Young as a source for learning more about MVC. The books has some mathematics but also a fair number of examples. What you are interested in falls under the nomenclature of "decomposition of the T^2 signal".
Within JMP I'd do all of the following:
1. Invoke the change point detection option to find 'when' something shifted. The change point detector only finds the first incidence in time...but it's a start.
2. To Karen's point, if you have a reasonably small number of original variables, try a multivariate scatter plot path. You'll get this through the change point detection option as well.
3. Make sure to use JMP's interactive linking between JMP reports and the original data table by clicking/highlighting interesting points on the various charts and graphs you will generate. This will help you find those interesting observations.
Outside of JMP:
4. Make sure you engage process knowledge experts with domain knowledge of the process and ask the probing questions..."What might have changed?", "What do we know with certainty did NOT change?", "Are there other process factors we should look at to provide hints?", "Are there suspected noise/nuisance variables which may have crept into the process?", "Is it possible we're observing measurement system variation and not true process variation?...if 'No' how do we know?" To often in industry I've been bit by this issue because people often blindly trust the measurement system as gospel...and there is no process monitoring in place for the measurement system...remember any observed process/product value variation is always the sum of true process/product variation AND measurement system variation. The last thing you want to do is react to measurement system variation.