There is presently not an easy way to migrate a subdomain from being handled by a different provider to being handled by Short.io without downtime because the DNS records need to change first, then an SSL certificate is issued after. In the time in between, the SSL will be broken because there will be no valid certificate and users will get a security warning.
This can be worked around by creating and uploading your own certificate before migrating, but that is a manual process and then you have to manage the certificate renewal yourself.
Providing the ability to manually verify ownership of domains/subdomains (and to automatically generate certificates in this scenario) would alleviate this problem. LetsEncrypt allows for domain ownership verification through TXT records and through email, so that could be an option.
In addition, if Short.io would verify wildcard subdomains, that would also allow for a zero downtime transition -- a wildcard subdomain handler could be created and the SSL certificate could be issued, and then specific subdomain DNS record that pointed to the old provider could be deleted, resulting in the wildcard DNS record taking over.