Francis of Assisi visited the Sultan of Egypt, according to Jacques de Vitry, the Bishop of Acre in the crusader kingdom of Jerusalem. Jacques saw the Franciscans in the crusader camp in Egypt during the Fifth Crusade, and was initially was suspicious of them, because they were a new monastic order that rejected the usual monastic traditions. But he eventually approved of their attempts to preach to Muslims:

“We have seen the founder and master of this Order, Brother Francis, a simple, uneducated man beloved by God and man, whom all the others obey as their highest superior. He was so moved by spiritual fervor and exhilaration that, after he reached the army of Christians before Damietta in Egypt, he boldly set out for the camp of the Sultan of Egypt, fortified only with the shield of faith. When the Saracens captured him on the road, he said: ‘I am a Christian. Take me to your master.’ They dragged him before the Sultan. When that cruel beast saw Francis, he recognized him as a man of God and changed his attitude into one of gentleness, and for some days he listened very attentively to Francis as he preached the faith of Christ to him and his followers. But ultimately, fearing that some of his soldiers would be converted to the Lord by the efficacy of his words and pass over to the Christian army, he ordered that Francis be returned to our camp with all reverence and security. At the end he said to Francis: ‘Pray for me, that God may deign to reveal to me the law and the faith which is more pleasing to Him.’ In fact, the Saracens willingly listen to all these Lesser Brothers when they preach about faith in Christ and the Gospel teaching, but only as long as in their preaching they do not speak against Muhammad as a liar and an evil man. When they do speak in such a manner, the Saracens irreverently put them to the lash and savagely expel them from their cities; they would kill them, if God did not miraculously protect them.” (Tolan, 19-20)

There are some unbelievable things in Jacques’ account - especially that the Sultan secretly wanted to convert - but otherwise, apparently Francis really was there and he really did meet the Sultan. And Jacques is certainly correct that Muslims were offended when Christian preachers insulted Muhammad, but he’s wildly optimistic about their fate. Franciscans, and the other order of preachers, the Dominicans, were certainly sometimes executed in Muslim lands. Some of them probably intended to get themselves killed all along, because then they'd be considered martyrs; in fact a few Franciscans were executed in Morocco around the same time Francis was in Egypt. Insulting Muhammad as a false prophet, or an idolater, or even Satan himself, was a sure way of getting your head chopped off.

As for the Dominicans, they preached all over the medieval world, even further than the Franciscans. Dominican missionaries even travelled to the Mongol court in China, such as William of Rubruck in 1254. They didn’t have much more success than the Franciscans though. Medieval Muslims generally thought Christianity was a defective form of Islam, so why would anyone convert? The Franciscans and Dominicans tried preaching to non-Latin Catholic Christians too, and there certainly were a lot of them - Greek Orthodox, Syrian Orthodox, Coptic, Armenian, Georgian, and Oriental Orthodox (or “Nestorian”) Christians were all missionary targets, but why should they convert to some other form of Christianity, especially one that considered them heretics? Latin Catholicism was the heretical sect, in their eyes. Preaching to the Mongols didn’t work out so well either. Some Mongols were already Christian, or Muslim, or Buddhist - but for the Mongols, whatever religion they were, they were the most powerful empire in the world, they had conquered most of Asia and some of Europe, so clearly, European Christians should be submitting to them, not the other way around.

I will have to leave the topic of Islamic proselytization to someone else. But Christians definitely did preach throughout the Muslim world. At best they were received with indifference at best, and at worst they were executed.

Sources:

John V. Tolan, Saint Francis and the Sultan: The Curious History of a Christian-Muslim Encounter (Oxford University Press, 2009)

Peter Jackson, The Mongols and the West (Routledge, 2005)