Bitcoin addresses are pseudonymous, in other words they aren't linked to anybody's ID without that user somehow linking the address to their ID (for example by withdrawing from an exchange service where they have provided their ID to the exchange).

There are various known tricks/tools to reveal IP addresses of transactions and to link various addresses to a wallet if they are all in the same actual wallet. Each have varying degrees of success. But at the end of the day if an address is used only once and isn't part of a wallet with other addresses in it, plus you use tools like Tor and mixing services it would be very difficult to link future addresses to the source address while still knowing for certain that the addresses belong to the same owner. Potentially impossible depending on the tools/services used. I'm not saying Bitcoin is anonymous, it's not at all, the entire transaction history of every address is entirely visible to every man, and his dog. But it is pseudonymous if used correctly and that's what is important for your question. The funds can be traced through the addresses, but tying the owner to those addresses is what's important.

For example, note that this is not advice, but if you sent your identifiable purchased funds to an exchange, gambling site, dark net site, or other site that does not require ID verification. Then withdraw those funds in multiple transactions to different addresses in different wallets, using Tor or similar perhaps. Then those funds would be indistinguishable from the source funds in most cases. But, do your research if extreme privacy is important to you. Most of us are happy just pushing the funds through a simple mixer for a little bit of semi-privacy.

If done correctly, then no, the offline-wallet address and the final address would be unlinkable (if considering only information recorded in the blockchain).

However, in order to do this correctly, there are a few points to be aware of and take care of:

1. If the amount you mix is some arbitrary amount X, and X is just an arbitrary amount, say 345.9372BTC, then it is easy to see the offline address and the final address have the same distinguishable amount, and link them -- For this reason, mixer services accept predefined denominations, e.g. 1BTC, and you should transfer each "chunk" to a different address.
2. Even if you use multiple target addresses, if you later transfer, in a single transaction, from those multiple addresses, it can be inferred that all those addresses are owned by the same owner. Under some circumstances, a link can be suspected to exist between those addresses and the pre-mixing address -- For this reason, it is recommended to carry out multiple mixing iterations, instead of just a single mixing.