Alexei Moskalyov, 54, had fled house arrest in Russia before a court handed him a two-year sentence in absentia for "discrediting" the Russian army.
Russian and Belarusian state media said he was being held "upon request of the Russian police."
Last year, Moskalyov's now-13-year-old daughter, Maria, drew a sketch at school of missiles next to a Russian flag flying toward a woman and child standing by a Ukrainian flag.
Memorial, the outlawed Russian rights group that won the 2022 Nobel Peace Prize, has declared Moskalyov a political prisoner.
What happened to Maria?
The girl's school had contacted the police. Maria was questioned and Moskalyov was fined
for earlier critical comments on social media, according to his lawyer and supporters.
The pair were separated as the daughter was taken to a local "rehabilitation center" for minors.
A lawyer representing the single father has said it was "difficult to predict" what would happen to his daughter.
A Russian court is expected to consider the authorities' petition to restrict Moskalyov's parental rights next week.
Kremlin spokesman Dmitry Peskov has defended Moskalyov's sentencing and described his parenting as "deplorable."
