Russian Wikipedia claims this:
Together with his brothers he was involved in revolutionary activities and was arrested at the age of 12. He was forced to emigrate with his family to Switzerland .
And that isn't referenced directly but apparently based on the following account:
At 12, Boris became a revolutionary. Together with his older brothers, he made bombs for the participants in the 1905 uprising. The Belousov brothers were arrested and sentenced to exile or emigration. The family was forced to emigrate.
— N Gorky: "The Tale of the Chemist Belousov, Who Made the Liquid Clock", Science and Life. 2011. No. 2. (Горькавый Н. Сказка о химике Белоусове, который изготовил жидкие часы // Наука и жизнь. 2011. № 2.)
Seems in need of a caveat:
The generation to which Boris Pavlovich Belousov (1893–1976) belonged has almost disappeared. The archives hold only a few documents about his life and basically secret research. This article is a brief biography of Belousov and attempts to reconstruct what lay behind his famous discovery of the oscillatory homogeneous chemical reaction named after him.
— Alexander Pechenkin: "B P Belousov and his reaction", Journal of Biosciences, Perspectives Volume 34 Issue 3 September 2009. pp 365–371. (PDF)
So the veracity of the following tale should be enjoyed with some salt:
The eldest, Alexander, 17, already a revolutionary. The plans are fascinating: to blow up, to shoot, to hide. He's soaked in Marx, studying him hard, and "Capital", of course, has already read it. And this book is wonderful, the highest literary merit of emotional, artistic and, of course, scientific. Brilliant romantic thought that has conquered so many.
Sasha Belousov, inspired by the idea of world justice, found a wonderful audience in his brothers, all involved in the revolutionary work, including 12-year-old Boris. And the revolutionary work, as everyone knows, is connected with chemistry. Chemistry is the best science for overthrowing the existing system teaches how to make bombs. The laboratory was built right in the attic of a Moscow house on Malaya Polyanka. The brothers were really passionate. Making bombs at the age of 12 is a pleasure! It's also a pleasure to test them! And so Mom wouldn't know! Boris Belousov's childhood is a dream (for Tom Sawyer and Huckleberry Finn).
Mobby Alexandrovna, daughter of Alexander Pavlovich Belousov, told me about the childhood and youth of my brothers. She was born on a steamboat between Japan and the United States when her parents were fulfilling a party mandate to raise funds in America and other countries for the future revolution. A wonderful woman and her son, Boris Rafaelovich Smirnov, became the "key" to the whole story.
In 1905, during the 1st Russian Revolution, Sasha Belousov, associated with the top of the Bolshevik faction, led a brigade of militants. When the revolution was crushed, Alexander managed to escape. Sergei was arrested first, he named after some party member who was more valuable to the cause than this boy. And he died in Siberia.
Apart from Sergei, two more brothers Vladimir (14 years) and Boris (12 years) were arrested. And he was sent to prison for particularly important criminals. Alexander was arrested a year later, but managed to escape from the Siberian exile.
Boris in those years still slept in a hug with a big teddy bear. The gendarmes had to bring him to prison for a young revolutionary.
— Проекты / Биология: Симон Шноль и его книга "Герои и злодеи российской науки" / Б.П.Белоусов и его колебательная реакция
Repeated as 'family history' here.
So, in 1905, 12-year-old Muscovite Boris Belousov and his older brothers were imprisoned for the production of explosives. Grenades were used by the guerrillas and vigilantes in the rebellious Presnya. To this day, this workshop on chemistry would be called preparation for terrorist acts. The guys were doing unsafe (in all senses) experiments in the attic of a large house where Belousov family lived. The family was not poor. Father worked as a bank employee.
Four of the five Belousov brothers were involved. The older, seventeen-year-old Alexander, who actually "propagated" the brothers for revolutionary activities, managed to escape from the police. Sixteen-year-old Sergei showed heroism: when arrested, he called himself a stranger's name. The comrade, whom he had thus shielded from arrest, was more needed by the revolutionary cause than the boy who followed him at the stage and eventually died in Siberia. Minors Vladimir and Boris Belousov were also to be sent to Siberia. But the gendarmes offered the mothers of the young revolutionaries an easier choice: to emigrate. The family went to Switzerland.
— Жизнь замечательных имен : Короткие истории о вещах и о людях, давших им свое имя хирург: Реакция Белоусова - Жаботинского May 23rd, 2012, 12:03 am
A biographical memory detail:
— Yes, I was in Switzerland together with my father, Alexander Pavlovich Belousov, a member of the military-technical bureau organized in 1905 by the Moscow bureau of the RSDLP to prepare an armed uprising, as well as with his brothers Vladimir and Boris. […]
Then we moved to Zurich. There lived a lot of Bolsheviks, friends of my father, his brothers studied there. It should be specified that my father was twenty-three then, Vladimir was nineteen, and Boris was seventeen. The revolution in which they participated took place five years earlier. Nevertheless, fourteen-year-old Vladimir, and twelve-year-old Boris really worked in the workshop, secretly arranged in the attic of the maternity hospital, were making bombs for Krasnaya Presnya. There was also Sergei, he was sixteen years old. In early 1906, the workshop was found, the guys were arrested. Such a detail: when they were placed in a cell, one was found under the cape of a teddy bear. Despite the years, the children were punished to the fullest extent. Sergei was exiled, he had not returned from Siberia. Vladimir and Boris were expelled from a commercial school with a wolf's ticket and sentenced to be expelled from Moscow. Both were in poor health. Their mother decided to send them not to the village, but to Switzerland.
— В.ПОЛИЩУК – НА ОБЩИХ ОСНОВАНИЯХ – рассказы о судьбе одной химической реакции и её создателя Взято из сборника "Пути в незнаемое", М., 1985г., (№18).
In this Russian article the legend of interest in chemistry from brotherly bomb building is repeated directly as well:
It is known that after the suppression of the 1905 revolution, their family was forced to emigrate to Switzerland because of the participation of their elder brother, Alexander Boris. There, Boris found himself surrounded by a Bolshevik colony. Fascinated by his brother's idea of building a bomb for his revolutionary work, Boris began to actively study chemistry in Zurich.
— L. L. Prilepskaya & E. Yu. Starikova: "К 60-Летию Открытия Колебательных Реакций" (For The 60Th Anniversary of the Discovery of Vibrational Reactions), Chemical Technology 111, УДК 544.431.8(09)
That said, it may be my poor search skills and 'slow Russian', but it looks a bit suspiscious to me that even the sources in Russian as listed by Russian search engines seem to almost always copy large parts of that story – if not perpetually verbatim – which always read like from a single well… A proper biography is apparently absent.