Under worldwide rules, Ethiopians are leading the investigation but France's BEA will conduct black box analysis as an adviser.
USA pilots have complained about the 737 Max, saying its systems have limited their control, according to a confidential US government report.
All 157 people on board - 149 passengers and eight crew - died in the crash.
Overnight, the United States' Federal Aviation Authority signalled the grounding of all 737 Max aircraft operating in American airspace.
On Wednesday, Ethiopian's CEO hinted in an interview that the black box was due to be taken to Europe for analysis given that Ethiopia lacked that expertise.
In the meantime, Caribbean Airlines is doing all that it reasonably can, to accommodate on its aircraft, those passengers of other air carriers who may be affected by the grounding of the Boeing 737 MAX-8.
None of the nation's carriers have the Max 8 or Max 9 in their fleets, the agency said.
The low-priced carrier, a launch customer of the MAX 8 in 2017, said it was servicing those flights with available 737-700s and 737-800s.
The airport is storing 10 of the aircraft and says it has the flexibility to store more.
The company continues its efforts "to understand the cause of the accidents in partnership with the investigators, deploy safety enhancements and help ensure this does not happen again", he said in a statement.
By the end of January, Boeing had delivered 350 of the Max 8 models.
Since the Indonesian crash, investigators have focused on the aircraft's anti-stall system that dips the plane's nose down to prevent it from stalling.
The accounts of the recent crashes were echoed in concerns registered by USA pilots on how the MAX 8 behaves.
It could also look at Grant County International Airport in central Washington, where Boeing regularly tests aircraft. Aviation experts say other technical issues or human error can not be discounted.
Boeing 737 MAX 8 and 9 planes will remain grounded for "weeks" at a minimum until a software upgrade could be tested and installed in all of the planes, USA lawmakers said on Thursday.
Andrew Hunter, a defence industry expert at the Center for Strategic and International Studies in Washington, said that while Boeing and the FAA had good track records on addressing safety concerns, sometimes the combination of automated systems and humans did not work smoothly.
The delayed flight ET302 took off from Ethiopian capital Addis Ababa at 08:38 local time (05:38 GMT) and was bound for Kenyan capital Nairobi.