Including (but not limited to) the PCs having a bounty placed on their heads, being declared wanted men in the Kingdom, or having the Kings men or similar people coming after them, in the same way the FBI would come after you IRL?
How is that illogical or immersion breaking?
It is illogical and immersion-breaking to assume something that exists in real life (FBI raids) would apply in substantially the same manner in a fantasy world, and that it is the only direct and natural consequence of the adventurers killing multiple guards in a prison break.
First, it assumes that the lord actually cares enough about several dead guards to send anyone after the adventurers. Maybe he does, but maybe he doesn’t.
Second, even if he does, it assumes that posting a bounty is a good idea. A good lord may not want to encourage his citizens (most of whom are commoners) to throw their lives away pursuing people who are guaranteed to tear them to ribbons. Even if this isn’t a problem, he must weigh setting a bounty against using that money for the good of his people.
He also needs to have that money handy in liquid form. Many feudal lords were land-rich and cash-poor. They simply couldn’t afford a thousand gp bounty to hunt down guard-killers, without seriously impacting their rulership.
You also need to consider the bounty hunters’ side of the equation. A high level adventurer just isn’t going to be tempted unless the bounty is really high. And morever, he doesn’t actually know the level of the target. Is the target 3rd level or 20th level? All he knows is that the target killed several guards.
Finally, you also need to consider tactics. Unless the bounty hunter has access to mid- or high- level divination magic, heading out into the wilderness to try and find the characters doesn’t make any sense. The wilderness is too big, and with poachers, travellers, hermits, woodcutters, humanoids, there is no assurance that w hunter will find there target. It makes much more sense to wait until they arrive at a settlement and hit them there before they melt into the wilderness.
But all this is moot. Because you are not sending “consequences” after the party because they are the natural and logical consequences of what preceded. You are sending consequences because you don’t like the actions the players have taken. Trying to solve an out of game problem (a mismatch between your preferences and the party’s preferences) using an in-game solution is not likely to work.