I'm just asking. When an MSE account is temporarily suspended to cool down, then the expected outcome is that the user would start doing the same things on MO which led to the MSE suspension. This has the effect of MSE dumping problem users on MO. Is this in MO's best interest?
No. The sites are different, also the standards and context are somewhat different. I do not see any reason for any automatic mechanism (or even only an informal one, except perhaps exchange of general information among the moderators) for moderating decision on the one having effect on the other.
If and when a user does something on this site warranting a suspension on this site, they can and should be suspended on this site. If not, then not.
If a user is going from site to site and creating disruptions, we (the Stack Exchange) community team will get involved. We don't like to preemptively suspend users on similar sites because suspensions are designed specifically to get a certain kind of behavior that is disturbing to a certain community to stop.
An example on another topic would be programming. I'm user12345 and I'm constantly letting the user DarthHater get under my skin on Programming SE. I've been contacted about it a few times by the moderators but I can't sleep until I tell this clown hat how wrong he is.
There's no reason why I should be suspended from Stack Overflow, Code Review, Code Golf and Wordpress at the same time that I'm suspended at Programmers. I've caused no disruptions within those other communities, and the atmospheres are completely different. Plus, Darth never goes to those sites, he stays on Programmers.
If it gets to the point that a user just leaves misery in their wake, it's time for someone from the community team have a chat with them, and possibly take additional administrative action to ensure the disruptions stop. That's what we're paid for, and we're always around if you need us. Well, usually almost certainly sometimes often somewhat always!
Of course the moderators can take into account that a user has been a problem also in some other online community. I guess this has already happened a few times, think of "Oh, this guy is a crank, he's well known on sci.math".
On the other hand I don't think math.SE should play a special role here, that is, I don't think disruptive behavior on math.SE should be considered differently from disruptive behavior on (say) sci.math as far as our moderators are concerned. For one, I have not had complete faith in all of the moderators on math.SE, and I would not like them to automatically have the power to influence decisions on MO even indirectly.
I don't think it makes sense to have an automatic suspension policy, however I do think the moderators should let their decisions be informed by all their knowledg about a user, including their m.SE behavior. In particular, if someone has already been warned and suspended for certain behaviors at m.SE and then they try the same thing here, I'm in favor of the moderators skipping the warning phase and going straight to a suspension.
I would say that the suspension on one site should not imply the automatic suspension everywhere, though moderators should be able to do it in the extreme cases manually (anyone with a temper worse than that of St. Francis can get involved into some local flame war, etc. and some fora may be more encouraging in this respect than others), but the suspension on two fora would usually be a sufficient evidence of the "inherent trolling inclinations" and "premeditated disruptive behavior" to warrant the automatic global suspension.