Travellers have flown into Ecuador without a carnet, but in the last year or so it has been very difficult without a carnet.
Legally it is required, and some travellers have had to wait for up to six weeks for a carnet to arrive from their home countries in order to clear customs. Other travellers (only a VERY few) have managed to enter without a carnet, but they spoke perfect Spanish, knew how the system works, and who to "convince".

Plenty of experience in Latin America, the right attitude and a "generous" nature
may be sufficient if you want to do this.
Colombia is possible, but the political/querilla situation is risky. A travellers has been kidnapped, but released - he was an Ecuadorian. Europeans and North Americans may not be so lucky. Ride only during the day, major highways, and straight out of Colombia is the best recommendation. The people are generally very good and very friendly, but it is possible to have problems. I was verbally assaulted by one farmer, but other Colombians held him away while advising me to leave the area.
Contact our Colombian Communities just before arrival for up to the minute advice.
Best may be to simply fly into Peru anyway, as Ecuador has been experiencing considerable unrest recently anyway.
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Grant Johnson
Share the Dream!
at:
www.HorizonsUnlimited.com