The section below contains statements Apple used to have on its page versions early 2017. Some of these no longer appear on those pages and I don't know if it is because Apple is changing some of its policies or if it simply is omitting policy statements from its pages that people won't like to see and they don't want to scare off customers:
Download your past purchases - https://support.apple.com/HT201272 - "Find your past purchases in the iTunes Store, App Store, iBooks Store, and Mac App Store on your iPhone, iPad, iPod touch, Mac, PC, or Apple TV" "With iTunes in the Cloud, you can access your past purchases from the iTunes Store, App Store, iBooks Store, and Mac App Store." - enabled with iTunes 10.3 and newer; not all media formats are available in all countries (see: iTunes in the Cloud, iTunes Match, and iTunes Radio availability by country - https://support.apple.com/HT204632); apps, books, audiobooks, music, t.v. shows, and movies (some - not all studios have permitted this). Downloading previously purchased movies and TV shows requires iTunes 10.6 or later. Discontinued items not available. For items not included in the iCloud list (e.g., ringtones, alert tones, rental movies), or locations or computer systems where iCloud is not (yet?) available, you only get one download per fee paid. "After you download your purchases, back up your iTunes library. Previous purchases might be unavailable if they've been refunded or aren't on the iTunes Store." (https://support.apple.com/HT201272). And: http://www.apple.com/legal/internet-services/itunes/us/terms.html#SERVICE - "It is your responsibility not to lose, destroy, or damage Content once downloaded. We encourage you to back up your Content regularly." ... "Content may not be available for Redownload if that Content is no longer offered on our Services."
If an item get pulled from the store, or it gets re-branded, it may not be available for re-download.
Also thoroughly read all the tips provided in the link in an earlier post. Are you using the same Apple ID? You have not changed country?
Why not just transfer items from your old computer to the new one?