So, a request for www.example.com/thispage on the old site returned 404 and now you redirect those requests to the new site?
Yes. Google will note the redirect and schedule the new URL for spidering at a later date.
Those requests will of course return 404.
The solution was to not have redirected requests that previously returned 404 on the old site.
It's too late to do anything now. Google will request the URLs on both sites forever.
There's nothing else to do other than ignore the list of 404 errors. The list is there in case you need to bring a page back that you didn't know was missing.