How to seamlessly transfer a domain (avoid downtime because the previous registrar's nameservers stopped serving DNS records)

Well, should have checked the NameCheap knowledgebase, first: How to transfer a domain into Namecheap without a huge downtime?

NameCheap offers a FreeDNS service so their name servers can start handling DNS requests before a transfer. I suppose it would keep working when transferring away from NameCheap, too.