This is normal business. Search engines are notoriously slow. The reason for this is simple- the Internet is flippin' HUGE!!
The best thing to do with the least amount of work is to let the 404's happen. Do not stop this process. This is the primary way that search engines know that a page is gone.
The very best thing to do is to issue a 410 error if you can. But that can take work that is not always easy to do.
The difference between a 404 (temporarily gone) and a 410 (gone) is that the 410 is immediate whereas the 404 is not. With the 404, Google for example, will retry the page several times over a period of time before determining that the page is truly gone. This is the standard way of course, but not as fast as a 410.
You do not want to stop or block this process. Otherwise, the search engines cannot know these pages have been removed.
If you have bots accessing your site that you do not want or appreciate, you can block those of course and you may want to for a period at least.