Like if someone did a horrible crime then escaped to a neighboring kingdom were they pretty much Scott free? Or could the neighboring kingdom be after them too or perhaps the first kingdom would send a troop of knights to bring them to justice. And if such a thing happened. Were these clandestine. Or was this a thing that was a breach of the other kingdoms sovereignty.
Or did all the above happen and it was just depending on the circumstances?
Short answer: no. But let me develop it a bit more with a famous case from Spain, though not from the Middle Ages, but a bit later.
Juan de Escobedo was the secretary of John of Austria. He got murdered in Madrid in 1578 when he was coming back to palace after a meeting with the princess of Eboli. The royal secretary Antonio Pérez was involved in the murder. He fell out of grace with King Philip II, so he decided to flee. He fled Madrid disguised as a woman and went to Aragon.
The justice of the Kingdom of Castile could not do anything against him, as he was in a different Kingdom. Pérez, being of Aragonese ancestry, plead for asylum before the Justicia de Aragón, which was granted. Antonio Pérez had been tried in Madrid, in absentia, and sentenced to death. King Philip pleaded to the Justicia to send Pérez to Castile for crimes of high treason and trafficking with state secrets. As the justice in Aragon was mighty slow, and king Philip was getting fed up, he had the Inquisition intervene, fabricating an accusation of heresy against Antonio Pérez. The Inquisition got him, but he was soon liberated by an angry mob claiming Antonio Pérez's inquisitorial trial was against the laws of Aragon. Finally, Antonio Pérez managed to flee Aragon and found refuge in Béarn, then in Paris, under the protection of the Kings of France.
There were many more cases of someone just fleeing, thus avoiding justice, most notably Juan de Valdés, in the early XVI century, who fled to Naples after being accused of heresy.
Another case, but this one is actually from the Middle Ages, would be Enrique de Trastámara, who had fled Castile, persecuted by his half-brother king Pedro I for treason. He found refuge in France, where he established himself as a mercenary while plotting to come back to Castile and take the throne.
Sources:
Escolar, Ignacio y Arsenio, El justiciero cruel, Madrid: Península, 2012.
Marañón, Gregorio, Los procesos de Castilla contra Antonio Pérez, Madrid: Espasa-Calpe, 1970.