Fully 4,934 people crossed into Hungary directly from Ukraine on Sunday, while another 8,781 who crossed from Romania said they had come from Ukraine, the national police headquarters said.
Police issued temporary residence permits valid for 30 days to 393 people, the police website said on Saturday. Holders of such permits must contact a local immigration office near their place of residence within 30 days to apply for permanent documents, it added.
Budapest police received 1,225 refugees, 415 children among them, by train, according to the Budapest police website.
Refugees arriving on special train services at Kőbánya felső railway station, in eastern Budapest, were taken by bus to the BOK sports and events centre serving as a humanitarian transit point.
The authorities transported 88 people, including 53 children, to accommodation in the capital and outlying areas.
The police have contracted 31 Ukrainian interpreters, and they are working closely with staff of the disaster management authority, the immigration office, the Budapest metropolitan council, as well as numerous aid groups, the statement said.