No, file copy via SMB does not send any cacheable information in the first place – both authentication mechanisms used by SMB are built to only send temporary proofs and not your actual password. (SMB generally uses Kerberos in Active Directory environments, NTLM in other cases.)
Kerberos is a ticket-based, pre-shared secret protocol. With Kerberos, the server only receives a ticket that was issued for that specific server and cannot be used to acquire more tickets – neither your password nor your initial ticket (the TGT) are ever sent. (The ticket itself is verified offline, by decrypting it using the server's "machine account" password.)
NTLM is a challenge-response protocol, in which the server never receives the original "NT password hash" but only a version that's hashed again with connection-specific random data. In order to verify the response, the server needs to either already know the NT hash (in which case caching it is moot), or to forward it to a domain controller which does (in which case the server never learns the NT hash).
In both cases, the server only receives derived credentials which are useless to cache. (LSASS only caches initial credentials – its purpose is to derive those temporary credentials without actually revealing the initial ones. Mimikatz finds ways to extract them by force, but that's not normal operation.)
Also, the server doesn't need your password to check file permissions – it only needs to know your user SID & group memberships, which it'll obtain from the Kerberos ticket or by querying the DC.
RDP and WinRM are a different case; they're interactive access protocols, so they deliberately have a way to forward (delegate) your initial credentials to the remote machine, although it's optional and controllable via GPO. That's to allow the user to further access file shares and other things while connected to the server.
(Also, RDP carries a lot of legacy – although it can use Kerberos or NTLM nowadays via NLA, the older non-NLA mode simply involves sending your actual password to the server and indeed simulating a local interactive logon, with caching and all.)