Speculations pre-full-release are always silly, which is also why they are fun. [...]
Pre-release speculations are one honking great fun---let's do more of those!
In the Prolog, Eirin says,
"If they're making a move, no humans will possibly be able to compete with them.
But if we stand against them, we'll be playing right into their hands.
Moreover, the youkai of Earth likely can't even interfere to begin with."
The passage reads as if
they are luring Eientei (Eirin and Kaguya) out, by forcing their hand as the only ones capable of resistance. This is compounded by the former's notions that youkai (likely) can't help, nor can humans fend the invaders off. (This seems to mirror their predicament at the beginning of IN, when they had to deal with matters themselves. Of course, Eirin---with her genius and preparedness---took the way out by enabling select humans to do so.)
We learn in Bougetsushou that the Lunar Capital are still looking for Eientei, the responsibility of which falls on the Watatsukis. It's possible that, in addition to purifying the Earth, the invasion is a means of capturing them. It seems like taking them to the Capital holds a purpose, despite the Earth already being an impure prison. (Close to the mad person, perhaps? Research into obtaining absolute immortality? The drinkers of the Elixir aren't impure?)
Having written the above, I realised that the Earth purification would lead to the same condition, i.e., Eientei would be within reach of the Capital. Instead of bringing them to purity, they bring purity to them.
Regarding the Watatsukis, Toyohime states that she and her sister are "the most closely-related" to Earth matters, so they would have some involvement with recent events. But their stance has always been defensive, and they expressed no desire in capturing their master and the princess, if the latter is indeed one of the goals. Story-wise, either they won't put much effort into it, like what others said, or they were stopped from interfering, e.g., made powerless, stripped from their titles and duties.
The link between ULiL and LoLK appears to be the ball from the Lunar Capital, which connects a place to presumably the latter (see Clarste's ULiL plot summary; contains spoilers). The chaos in the Capital and perhaps the invasion are caused by one lunatic, which hints at this person's very high rank and status. According to Toyohime, when Earth discovered that reality is based on probabilities and gained the potential to reach the Capital, she named one person who was particularly affected. Is the mad person perhaps their Lord Tsukuyomi?
Tsukuyomi is the one who realised that impurity drains life on the Earth, and left with his trusted relatives for the Moon. Coupled with the threat of further defilement, rectifying what originally drove him out sounds like a legitimate move. Maybe the title
Lunatic Kingdom is a hint in this direction? (I can imagine his affairs becoming worse when he no longer had Eirin's intellect to rely on. And I'm not saying we'll definitely get a male boss; there's always the first time, but ZUN can just throw in a twist and have us fight someone else instead, say his sister, Amaterasu.)
For more details, CiLR's third chapter discusses significant topics and seems very relevant IMHO.
@HalfGrand: Doremy and sweets for me. :V If I were to go by my theory, there is another connection: Amaterasu became angry with Tsukuyomi when he killed Uke Mochi, the goddess of food.