There are multiple instances in which the Mongols and Manchus sought to influence the selection of the Dalai Lama. The Khoshot Mongols led by Lazang Khan invaded Tibet and captured Lhasa in 1706. Lazang Khan removed the sixth Dalai Lama from power and appointed his own Dalai Lama. This angered the Tibetans, who were still loyal to the previous Dalai Lama. The former sixth Dalai Lama was expelled and sent to the court of the Qing emperor Kangxi, but died mysteriously en route. It's possible that he was assassinated under Lazang's orders, but he may have just passed away from an illness. A child born in Kham was identified as the true reincarnation of the sixth Dalai Lama. The Qing army expelled the Mongols from Lhasa in 1720 and the seventh Dalai Lama was brought to Potala Palace, replacing the Dalai Lama installed by Lazang.

During the Qing dynasty, the Manchu emperor tried to intervene in the selection process of the Dalai Lama by instituting the golden urn lottery, beginning with Emperor Qianlong in 1792. Ostensibly, the golden urn served to make the selection of the Dalai Lama less nepotistic by randomizing the process (it's no coincidence that Dalai Lamas tended to come from powerful noble families). Qianlong blamed the Gurkha invasion of Tibet on the hereditary selection of lamas. However, the golden urn came to symbolize the Qing government's desire to exert more authority over Tibetan Buddhism,¹ so it was seldom used by the Tibetans, who preferred more traditional methods for selecting the Dalai Lama. According to Yale historian Peter Perdue:

The Golden Urn procedure was designed to break this grip by establishing the Tibetan Buddhist hierarchy as a separate, parallel one under Qing official control. It was, in effect, another version of the time-honored imperial practice of "setting barbarians against barbarians," an attempt to split the Tibetan Buddhist clerics from the Mongolian Khans. Even though Qianlong regarded the belief in khubilghan reincarnations as illogical and contrary to the principles of Buddhism, he found it a useful instrument for imposing control over the Mongols and Tibetans.

This is just one example of how the reincarnation of the Dalai Lama was subject to controversy even prior to the 20th century. The struggle for authority between local Tibetan leaders and the government based in Beijing is hardly a new phenomenon. So it's no surprise that a succession of mostly short-lived Dalai Lamas, from the 8th Dalai Lama to the 12th Dalai Lama, led to rumors that the Dalai Lamas had been poisoned by either Chinese representatives in Tibet or by Tibetans who felt the Dalai Lamas lacked legitimacy because the "wrong" Dalai Lama had been chosen.

These rumors are repeated by Thubten Jigme Norbu, the brother of the 14th Dalai Lama:

This move having failed, and the Tibetan people still remaining obstinately devoted to the Gyalwa Rinpoche [Dalai Lama] as their leader, the Chinese attempted to control the section of the Gyalwa Rinpoche. It is perhaps more than coincidence that between the seventh and the thirteenth holders of that office, only one reached his majority... The circumstances are such that it is very likely some, if not all, were poisoned, either by loyal Tibetans being Chinese-appointed impostors, or by the Chinese for not being properly manageable.

The veracity of these rumors have not gone uncontested, most notably by the 14th Dalai Lama:

"Yes," he said. "I don't accept that they were poisoned. It was just really carelessness. Of course, the possibility is there, but I feel that it was mainly negligence or foolishness that caused so many young Dalai Lamas to pass away. Look at the case of the Teenth, or the Eleventh. According to biography of his life, he was sick for a few months and coughing. To me, it sounds like tuberculosis. Among the elderly people who cared for the Dalai Lamas, they had great devotion, but they really didn't know how to take care of them.

¹But, as Dawa Norbu has pointed out, the golden urn was likely derived from a Tibetan tradition called tagdril that predates the Qing period.

Sources:

Peter Perdue's China Marches West: The Qing Conquest of Central Eurasia

Thomas Laird's The Story of Tibet: Conversations with the Dalai Lama

Thubten Jigme Norbu and Colin Turnbull's Tibet

Don't they roam the world, not just the country?

You might want to try /r/buddhism or /r/metabuddhism

Although HH Dalai Lama is the most famous "tulku" (reincarnate lama), there are tons of tulkus. The first lama to be recognized as a tulku was the Karmapa, the head of the Karma Kagyu sect. There is a "who's the real Karmapa" controversy at the moment, with two men recognized as the 17th Karmapa by different factions. https://en.wikipedia.org/wiki/Karmapa_controversy

There are a lot of examples of tulkus behaving badly. Chogyam Trungpa, the Oxford educated tulku who brought the teachings to the West and wrote a number of books, was infamous for drinking and sexual misconduct. https://en.wikipedia.org/wiki/Ch%C3%B6gyam_Trungpa#Controversies A few of his students left. Most stuck by him, including the famous Pema Chodron.

How do students rationalize bad behavior? They consider it teaching. Patrul Rinpoche's teacher Dho Khyentse was stinking drunk, smacked him around, gave him the finger and said, "How does such a thought enter your mind, you old dog?" Which is when Patrul Rinpoche recognized the nature of mind. Pema Chodron considers her teacher's bad behavior as a sort of koan.

u/budars got everything pretty much right. I'd like to add that the Sixth Dalai Lama is a perfect case study of "getting the 'wrong' Dalai Lama."

Essentially Tsangyang Gyatso was selected to follow the "Great Fifth" and ended up refusing the monks' vows at the great concern and alarm of the Ganden Phodrang (the monastic government of Tibet). There is no precedent to "recall" a Dalai Lama and plenty of tulkus (reincarnated masters) have chosen to simply not take their monastic vows and live the life of a layperson. Not only that, but Tsangyang Gyatso was something of a libertine, spending his days drinking, writing poetry, and having sex. This sort of behavior angered both the Tibetan monastic government and the Mongol military one and he was eventually imprisoned and went to China to summon military assistance but died along the way.*

Lazang Khan plucked a random monk and declared him the "true" Sixth Dalai Lama as Tsangyang Gyatso was selected by mistake. Ngawang Yeshe Gyatso (or so wikipedia claims. I'm in a different country and don't have my copy of The Fourteen Dalai Lamas by Glenn H. Mullin) never made the claim to be Dalai Lama himself, and was viewed as more of a regent until the Seventh Dalai Lama would arrive. He was never viewed as a Dalai Lama by most Tibetans and current theology states that Chenrizi (the bodhisattva of compassion) takes a thousand incarnations at any one time, and he put himself in front of Lhazang Khan when he was picking a "true" Dalai Lama as replacement.

The "theology" behind tulku selection and what happens if/when a tulku reveals s/he is not the true incarnation can be complicated at best. I suggest Secrets of the Vajra World by Reginald Ray which deals with this and plenty of other topics intensely detailing the Buddhism of the Himalayas.

*His death is still controversial. Some allege that he died en route as a part of a promise made to "never leave Tibet." The Dalai Lamas leaving Tibet is universally taken as an omen of destruction for the nation. So the most popular legend among Tibetans includes a ritual suicide by Tsangyang Gyatso in Amdo just before the border of Chinese territory.