How does certificate revocation work with intermed

Suppose a PKI hierarchy like below.

root CA ==> inter-1 CA ==> user-1
  \
   \======> inter-2 CA ==> user-2

My question is: does root CA also need to periodically download CRL from its children: inter-1 and inter-2?

Since user-1 and user-2 can authenticate each other, if user-2's certificate is revoked by inter-2, inter-2 should let root know and then propagate to inter-1 and user-1, right?

If so, it seems quite complicated. Is there any tool to use for managing the revocation logic? Thanks a lot.

No, revocation of certificate is not propagated across the CA tree. Each CA (root and intermediate in your case) is responsible of the publication of the CRL containing the list of only the revoked certificates that were issued by this CA.

An example:

Root CA publishes a CRL for the certificates issued by Root CA: inter-1 CA and inter-2 CA. Root CA is not aware of the user-1 and user-2 certificates or their revocation status.

inter-1 CA (resp inter-2 CA) publishes a CRL containing the list of revoked certificates issued by inter-1 CA (resp inter-2 CA) and only these certificates.

CRL Root CA   CRL inter-1 CA 
  ^             ^
  |             |
root CA ==> inter-1 CA ==> user-1
  |
  |           CRL inter-2 CA 
  |             ^
  \             |
   \======> inter-2 CA ==> user-2

if user-1 certificate is revoked, this certificate (actually its serial number) will only appear in the CRL published by inter-1 CA.

When someone wants to check the validity the user-1 certificate the process is as follows:

build the certificate chain between the certificate and a trusted CA: user-1 / inter-1 CA / root CA
fetch the CRL for the first certificate in the list
verify the signature of the CRL
check the status of the first certificate in the list against this CRL
if the status is not revoked, remove the certificate from the list and go to 2. otherwise fail
if the list contains only the trusted CA, check the chain of signature of the certificates (a certificate must be signed by the following certificate in the list)
if all signature have been checked and are valid, the user-1 certificate is valid.

Note that validating the CRL signature can trigger a validation of another certificate chain : i.e. this algorithm can be recursive. Actually the X.509 certificate validation algorithm is (very) complex and I just summarize the principles here.