They are among eight people still on the passenger list after dozens won legal cases and were removed.
A last-ditch attempt to block the flight altogether was rejected by the Court of Appeal on Monday, however.
The flight is likely to cost more than £500,000, but ministers say it will disrupt the business of traffickers.
As the archbishops of Canterbury and York joined opposition politicians in condemning the plan, a government spokesman acknowledged more last-minute legal challenges were expected but said “we will not be deterred” from starting the flights.
Pressing ahead with the policy of transporting asylum seekers to the east African nation would “break the business model of vile people smugglers” and “ultimately save lives”, a spokesman said.
The flight on Tuesday evening was originally due to carry dozens of passengers, but most succeeded in their individual appeals against deportation.
It was not clear exactly how many will leave on the flight: on Monday night, the Home Office said it was eight, while the charity Care4Calais said the number had fallen to seven.
In a letter to the Times, senior Church of England leaders described the plan as an “immoral policy that shames Britain”.
Signed by the archbishops of Canterbury and York and more than 20 other bishops who sit in the house of Lords, the letter said those being deported have had “no consideration of their asylum claim… or any attempt to understand their predicament”.
Tuesday’s flight is due to be the first in a five-year trial, in which some asylum seekers deemed to have entered the UK illegally are transported to Rwanda to claim refuge there.
They will get accommodation and support while the Rwandan government considers their application, and if they are successful they can stay in the east African country with up to five years’ access to education and support.
If their asylum claim is unsuccessful, they will be offered the chance to apply for other immigration routes, but could face deportation from Rwanda.
The letter, signed by the entire senior leadership of the Church of England, said those being sent to Rwanda have had no chance to reunite with family in Britain.
“Many are desperate people fleeing unspeakable horrors. These are people Jesus had in mind as he said when we offer hospitality to a stranger, we do it for him,” it says.
“We cannot offer asylum to everyone, but we must not outsource our ethical responsibilities, or discard international law – which protects the right to claim asylum.”
It’s not the first time the Archbishop of Canterbury has criticised the plan – he described it as “the opposite of the nature of God” in his Easter sermon.
At the time, the Home Office responded that the UK had a “proud history” of supporting those in need.
The BBC has contacted the Home Office for a response to the letter.