The "Sent unsolicited preacher" opinion modifier lasts 5 years, and prevents you from sending any more preachers, as well as giving them a -10 opinion penalty on you. It's easy enough to check if it's there - just hover over their opinion of you...
I had it stacked once, and the Ilkhan had 3 of my chaplains in his jail. I was able to reappoint and resend immediately. But this is different with the case when the khagan of the Golden Horde got worked up about a courtier of his getting converted.
Oh, and this is slightly off-topic, but when the Ilkhnate was wiped out by the GH and really stopped to exist (the former Ilkhan was reappointed as a simple chief to his last county after even it was taken away from him), the Mongol prisoner I held in jail switched to having me as his liege, since the Ilkhanate discontinued and the Ilkhan became unlanded for a second. He had like 23 Martial, Inspiring Leader and Cav Commander. He's currently Catholic and leading holy order stacks for me.
I'm not sure off the top of my head, but dude seems to have only positive traits, something of a True Christian Knight material. Almost married him to my doge's dad's young widow (3rd wife).
Also, GH has currently just DoW-ed me despite being at +100 opinion. Due to how things work, he's likely to be out of event troops and rather weak at the end of this, especially considering that as soon as his own declared war is over, I'm following up with a holy war for the biggest duchy he has conquered, which is also full of Muslims under his Tengri rule (who are Abrahamic, after all).