The bodies of the migrants -- two of whom were just 15 years old -- were discovered inside the sealed unit at a port near London in October 2019.
They had traveled in a truck from northern France to Belgium before crossing the Channel to Britain.
The 19 defendants in the French trial include people of Vietnamese, French, Chinese, Algerian, and Moroccan nationality.
Prosecutors described a vast network of illegal immigration presided over by the 19 defendants on trial for aiding the illegal entry, movement, or residence of foreigners in France as part of an organized gang.
They are also seeking convictions for criminal conspiracy, offenses punishable by 10 years' imprisonment.
Only four defendants -- known as "Tony", "Hoang", "Long" and "Thang" -- are on trial for manslaughter.
"The defendants prioritized profit, endangering the victims who saw them (the migrants) only as 'chickens' to be stacked. Some of the defendants cynically portray themselves as victims of the system," said prosecutor Alexis Liberge in his closing speech.
On Oct. 22, 2019, 31 men and eight women aged between 15 and 44, all from Vietnam, were loaded into a trailer in northern France and found dead the next day in an industrial estate east of London.
The disaster thrust the debate about people trafficking and migrant routes of entry into the spotlight.
Prosecutors are seeking nine to ten years imprisonment, along with fines and permanent bans from France, for Tony, Hoang, Long, and Thang.
The four other Vietnamese defendants -- two of whom were not present at the hearing and are considered to be fugitives -- will be sentenced to nine to ten years' imprisonment.
For the other defendants, the prosecutor sought acquittals on charges of criminal conspiracy, but convictions on the charges of aiding and abetting illegal entry, movement, and residence of foreigners in France as part of an organized gang.
The trial is due to end on Friday.
Two ringleaders of the operation -- one Romanian and one British -- were convicted at a trial in 2021 in Britain and sentenced to 27 and 20 years in prison.
Other suspects, drivers in the network, received 12 to 20 years, while a Belgian court handed a 15-year term to a Vietnamese man for heading the local cell of the gang.