Traffic was disrupted in some parts of the country as protesters tried to set up roadblocks using burning tyres.
Several passenger vehicles were also set on fire in various parts of Bangladesh in the run up to the protest.
A 60-year-old driver of a passenger bus was killed by smoke inhalation after people set fire to the vehicle in the city of Faridpur early on Tuesday.
Elsewhere, about 200 protesters briefly blocked traffic on the 6.2km (4 mile)-long Padma bridge across the Ganges, that connects the capital Dhaka with the south-western region.
Bangladeshi local media said dozens of crude bombs exploded in the past few days in Dhaka and elsewhere.
“Any attempt to worsen the law-and-order situation or create instability through any conspiracy within the country from across the border (India) will not be allowed to succeed,” Bangladesh Home Minister Salahuddin Ahmed said.
“Our supporters were not responsible for the violent incidents. They were carried out by unknown miscreants. We believe in peaceful protests,” a senior leader of the BCL, who is now in exile in a foreign country, said.
Hasina fled to neighbouring India two years ago following a massive uprising, bringing an end to her 15-year reign.
Opposition parties accused Hasina of jailing political rivals, supressing dissent and overseeing widespread human rights abuses. She always denied the charges.
The United Nations has estimated up to 1,400 people were killed, most by gunfire from security forces, during the anti-government protests in 2024.
The activities of the Awami League were banned soon after she fled the country and it was not allowed to contest the February general election.
A special Bangladeshi tribunal last year sentenced her to death in absentia after being found guilty of crimes against humanity.
She has denied all the charges, calling the trial “biased and politically motivated”.
Despite the death penalty threat, Hasina said she wanted to return to Bangladesh in December and requesteded the Bangladesh High Commission in New Delhi to issue her with a passport.
But the Bangladeshi government said the country would not provide a passport to her and called her “a terrorist”.
“She is a fugitive criminal. We have an active extradition treaty with India. We want her handed over,” Humaiun Kobir, Bangladesh’s junior foreign affairs minister, said.
“Should we treat her like an ordinary commercial traveller and tell her to come and visit her in-laws’ residence?” Kobir asked.
A state prosecutor in Bangladesh on Tuesday said the verdict would be executed if she came back, saying she had missed the opportunity to appeal.
Awami League leaders allege that in the past two years hundreds of supporters have been killed in mob violence and dozens have died in the custody of security forces.
Dhaka officials reject accusations that Awami League supporters are being systematically targeted.
