It's not a stunt. It's a message, and it's unfortunate. I don't like it, but I completely get it.
None of those (terrible) leaders up North care about the border until a bus shows up, and then they bitch about the governors of Texas, Florida and Arizona?
They should be bitching at the true cause! Maybe take a trip down to witness what's happening? Or just welcome them into your supposed sanctuary chitty with open arms?
....
In the case of Chicago, the mayor (for now) immediately had the last bus shipped to a conservative wealthy suburb - also without warning.
Hypocrisy.
If it's a message, it's not one aimed at the other side. Let's put it this way, if some folks in opposition to you tried to send a message that way, there's not much chance it would resonate, most likely it would harden you.
In another sense, you point out, this hasn't actually triggered much response of "Oh God, now these illegals are here." The complaint has been "Wow, that's a gross, weird stunt." In some cases, they have been welcomed (we'll get to Chicago later). It does nothing to change people's thoughts on the border. At all. And if it did, it wouldn't do so in two days.
Now, this is in some ways a message to the base. It gives the base some har hars. A chance to sort of jump the gun about the response and remember that we'll be fed "we'll fight for you" culture war slop going forward. And the other side will almost assuredly do the same, thus dividing us all for the mirthful fun of it.
The Chicago part is quite illustrative. Sending more illegals to Chicago is ... theater and little else. The city/region has plenty of illegal immigrants. Just about no one in that city is saying "Oh gosh, someone from the other side of the border, never seen one of them." (No one who's gonna see the new group anyway) It's just theater to give the base a chance to chuckle at a juvenile burn. The Mayor proved she's not above such juvenile things. She did her own stunt. And the only message it sent was that the person organizing that busing to the burbs is a real asshole (which she is!).