It is possible to add a searchspec to child integration components but some restrictions apply. you must also search on the parent.

Lets take the example of parent Accounts and child Contacts.

If you do a search on Account Name = ABC and contact Last Name = Smith it will first query all accounts whose name is ABC, then for each account it will filter the child contacts whose last name is Smith. If one of the acount records do not have a contact Last Name = Smith, the parent will still be in the output.If you only search on contact Last Name = Smith  without any search on the parent account it will still run and no errors will be seen. But the results may not be what you are looking for. It will bring back ALL accounts as you passed no search on accounts.
Then for each account it will bring contacts whose last name is Smith . If a parent account does not have a contact whose last name is Smith the parent account will appear in the result set anyway, without any children.

The important point here is it applies the searchspecs separately. First at the parent (even if no searchspec was passed to the parent) and then to each child of the returned parent records. Doc ID 523483.1 explains this in more detail, but it uses as examples the searchspecs '[Account.Updated] =' +Today() and [Contact.Updated]=' + Today()