Oh my God please, THIS, do THIS!
The idea of the Lich and/or a thrall kill squad spawning in any regular mission where assasins can spawn is AMAZING.
With all the gates to "opt in" to the creation of a Kuva Lich, the system should indeed be made into something that is showing up everywhere you go. So here are my thoughts to further add to this EXCELLENT kill squad suggestion.
The number and level of thralls spawned as a kill squad could be directly linked to the level of the Lich (mirroring the Lich's "influence" and amassed "power" over the solar system), and their probability of spawning should increase based on the Lich's rage system. Meaning, if your Lich owns 5 planets, of course he will have access to more, and higher level minions, compared to when he only owned one. And if he is absolutely pissed at you, you should expect him sending said minions every chance he gets to off you. This will also make murmur farm much more integrated with the rest of the game. Especially if said kill squads can literally spawn everywhere (with the exception of say Sanctuary and story quests). Imagine running a bounty on the Plains and a kill squad spawns inside your bounty target area. Wouldn't that be the whole point of having an a-hole after you all the time?
To further this idea, probably for future implementation: what if the way the Lich levels and takes over a new planet, is by showing up with a kill squad to a planet he doesn't currently own, and you fail to take him down for good? Meaning, yes, the level up would still be because you didn't have the proper sequence of requiem mods, but at least you failing to take them down "there and then" and them escaping and surviving will now make sense. Your enemy lived to fight another day, grew stronger, and now their influence spread on that planet as well. Now the Lich steals your stuff on that planet too, and the kill squads will be tougher.
As a side note, ignoring a thrall kill squad and/or a spawned in Lich in order to carry on and complete your objective SHOULD still be allowed and possible. However, doing so for a set amount of times on the same planet (let's say 4 times you ignore a thrall squad/platoon on the same planet) should also result on the Lich "conquering" said planet and leveling up. As in, you ignored the problem, now the Lich took over that planet and grew stronger.
It's essentially the same thing as now but
1) it would be more integrated with regular missions (lich missions on planets that are already taken over should remain as they are, for a direct path to the Lich, however with this the Liches and thralls will also be part of regular gameplay)
2) it would feel like the Lich "earned" their new territory. Sure, we would still fail to kill them just because we didn't know the correct answer to the "puzzle", but let's face it, if it only came down to damage any Tenno would down their Lich on first encounter with a skana. The Lich getting stabbed, then laughing it off and escaping means our stab triggered their evolution, their escape gave them a chance to amass more forces. Hence, level up and new planet owned.
3) The added ability for the Lich's forces to advance on and take over a planet because you were there and simply ignored them would also add to the whole "personal nemesis" thing. "I was there, I could have stopped them, but I chose not to because it was a hassle or I was underprepared. Well now a whole planet is under their control and I have nobody to blame but myself." That kind of thing.