Around 90 lorries from a Russian aid convoy crossed the border with Ukraine on Friday, heading towards Luhansk in eastern Ukraine.
Kyiv said it considers it a “direct invasion“ by Russia of Ukraine, but that it will not use any force against the convoy.
The International Red Cross said it was too dangerous for them to accompany the lorries as it could not be sure its workers would be safe.
A spokesperson said: “The Red Cross did not receive enough of a security guarantee to accompany the convoy. At the moment the Russian humanitarian aid convoy is on the move – the Red Cross is not participating and is not accompanying it.”
The around 280 vehicles carrying food and medicines had been stuck at the border for more than a week.
Moscow had accused Kyiv of deliberately holding up the delivery of humanitarian supplies.
The aid is intended to help the humanitarian crisis in the city of Luhansk.
Ukraine has previously claimed the convoy could be used to supply military equipment to the rebels.