How to use nmap through proxychains?

The support for proxy with nmap is very limited. Especially you cannot do any kind of ICMP (ping) or UDP scans, no SYN stealth scan, no OS detection etc. This means that the default nmap commands you are using will not work with a proxy and depending on the implementation will either fail or will bypass the proxy. You have to limit yourself to only the kind of scanning which is supported through proxies, i.e. simple TCP connections.

For more details about this see Nmap through proxy.

You have to use the -sT option -- the Connect() scan technique. Otherwise nmap will use the SYN method, canceling out proxychains. For you example, this would be

proxychains nmap -sT -v scanme.namp.org

And, by the way, proxychains-ng seems to cause more problems with nmap than the original version.