Yes puthujjanas struggle with the sensitivity to lust, just like sensitivity to sankharas, and they always try to salvage the vedanas of the usual 5 senses, like the puthujjanas who invented mahayana who keep claiming that there can be contact without craving, even for sensuality.
Then some puthujjanas try to express the dependent origination through time, which results in ''models'' with 1 life, 3 lives and a few others, but that's not the point of the DO.
Then they ask what is reborn, is there a state between death and birth. But that's still not the point.
For the DO, there is a famous sutta where the buddha question the Arhat Sariputta https://www.sariputta.com/sutta-pitaka/498-kalara-sutta/english
When it comes to the 5 senses, lust follows from contact and there is always lust as long as you are not in the fourth jhana. Lust is suspended in 4th jhana. most Puthujjanas struggle to get the citta into the samadhi of the 4th jhana, because all they know is sensuality and lack of citta in samadhi, which makes it easy for them to claim that they do not have lust. They claim that they do not have lust for some object they cognize, like some rock or twig in their garden, but they forget to claim that they still lust over other things that they cognize.
This obsession with caring about individual sensual objects and the individual tastes, likes and dislikes, is what make them claim that ''each person'' has a path to reach nibanna. They think that personal tastes matter, because they completely fail to see the big picture, that there is nothing to salvage when it comes to kama and that the only good relationship with kama to have is the one of upekkha. Kama is 100% dukkha. Vedana is 100% dukkha. There is not a tiny bit of rupa, of vendana, of jati, of karma, of anything conditioned that can be rescued form dukkha and converted into something non-dukkha.
Once you get your citta into the samadhi of the 4th jhana, you reach for the first time upekkha. This is not upekkha like in sensuality where you do not care about one particular object (again like a rock in your garden or particular tree in some forest), while still having plenty of lust for other objects (like plenty of other trees in the forest), but upekkha towards all the forest, all that is rupa, vedana and so on.
In terms of cessation the jhanas are expressed thus
Rahogataka Suttaṃ - Alone
"Then, bhikkhu, I have also taught the successive cessation of
For one who has attained the first jhāna sa eech has ceased.
For one who has attained the second jhāna hought and examination have
For one who has attained the third jhāna rapture has ceased.
For one who has attained the fourth jhāna in-breathing and
out-breathing have ceased.
For one who has attained the base of the infinity of space, the
perception of form has ceased.
For one who has attained the base of the infinity of consciousness,
the perception pertaining to the base of the infinity of space has
For one who has attained the base of nothingness, the perception
pertaining to the base of the infinity of consciousness has ceased.
For one who has attained the base of
neither-perception-nor-nonperception, the perception pertaining to the
base of nothingness has ceased.
For one who has attained the cessation of perception and feeling,
perception and feeling have ceased.
For a bhikkhu whose taints are destroyed, lust has ceased, hatred has
ceased, delusion has ceased.
When the citta is in the samadhi of the 4th jhana, what remains is the famous ''ignorance'', because the puthujjanas who do not have right view, sati, right resolve, and yet who manage to go into this jhana full of upekkha claim easily that this jhana is the end of the path.
This is what happened to the buddha with the famous legend that when he was a child he managed to get his citta into the samadhi of the first jhana. Fortunately, he was not stupid enough to claim he was awaken when his citta got out of this samadhi, but he did nothing with his first jhana.
Only decades later, once he wanted to end dukkha, and once he left all the wrong ways invented by the puthujjanas to get the citta into wrong samadhi (and yet that they claim is nibbana), the thoughts occurred to him that the good manasikharas are the ones of metta and nekkhama.
[at this point people today speculate whether he was a sotapanna, but the only explicit state mentioned is bodhisatta and what people call today ''buddhahood'' but sometimes he used a phrasing like ''when I was still not fully enlightened'' so perhaps at this point, he was a sotapanna, or just a lucky puthujjana who managed to see what happened when thoughts of nekkhama are cultured to the end, or perhaps what he called bodhisatta in some places include sotapanna...]
Then he says that even with what later on he will call right intention, meaning culturing thoughts of nekkhama and metta, he still failed to get the citta into right samadhi.
After this, he continued the right use of mano which is judging your skills at knowing good states from bad states and he continued to use mano rightly, meaning to see the danger of sensuality. after that he got his citta into samadhi, then his citta knew about karma-rebirth, then how to end dukkha.
Some puthujjanas ask how to ''stop desiring with desire'' and how to make nibanna the object of meditation or contemplation, and ''is it possible to crave nibanna''.
The answer is that it is not possible for puthujjanas to crave nibanna, because puthujjanas do not know what nibanna is and is not (you have to find out, by yourself, the meaning of the various statements of the buddha, because nibanna cannot be experienced vicariously). Even for non-puthujjanas, it is not possible to crave nibanna. Non puthujjanas go mostly by being repulsed by anything conditioned, and more precisely by what is disgusting, so typically kama and they strive for peace.
Puthujjanas always use mano to turn mano into papanca and the unhealthy way for puthujjanas to ''desire nibanna'' always is in fact to cling to what they fantasize as nibanna. It is part of the fantasy of puthujjanas that ''intellectual knowledge'' is indeed knowledge.
The desire to reach nibanna is typically generated by puthujjanas who live around non-puthujjanas, because puthujjanas love to mime, envy, idolize people who are allegedly better than them, and non-puthujjanas brainwash the puthujjanas with healthy statement and activities. But of course, when there is no arhat, it gets difficult for puthujjanas to desire nibanna.
What you still can do is to focus on sensual object that you do not care about and then use the mano properly by wondering if it would be good if you had the same relationship of ''not caring'' with the objects that you love, then with the objects that you hate. Use the mano to see the danger of having lust-hate over sensual objects. Use the mano to see how good you are at not lusting over sensual objects [typically food]. This is what ''guarding the senses'' is all about, having ''sati sampajanna'', ''being mindful''. When you are ''mindful'', you get to be sensitive to future lust and aversion, and with the memory of the statements of the buddha, you strive to keep this sati sampajanna (puthujjanas love to claim that having love and hate is good and not having them is bad).
This is how close puthujjanas can go to desiring and mediating on nibanna.