Finding the prime factors of $n'$ given the previous $n$ and its prime factors $p$ and $q$ Say I have a very large number $n$ = $pq$ where $p$ and $q$ are prime. 

A new number $n'$ is generated by increasing these prime factors. How can I find this new number's prime factors?
On one hand, what's asked in the question's title is finding the factorization of $n'$ given the factorization of "the previous" $n$ as $n=p\,q$. As far as we know, that can't be done for $n'$ of cryptographic interest, for a definition of "previous" implying a small increase in value.
That's part of surrogate factoring, which is to factorization what perpetual motion is to movement.

On a second hand, what's asked in the question is different: finding the factorization of $n'$ given that it is of the form $p'\,q'$ with $p'$ (resp. $q'$) a prime "generated by increasing" some known $p$ (resp. $q$). That makes the task easy, for a definition of "increasing" implying a small increase in value. All there is to do is explores the integers $p'$ above $p$ sequentially, and try if they divide $n'$, until hitting the factorization; it is not even necessary to check that $p'$ is prime.

On a third hand, a comment tells that "$n$ grows by a digit each round", which implies that at least one of $p$ or $q$ has changed a lot in value. If only one has changed a lot, we can try the above method on both $p$ and $q$. If both have changed, and we have no clue about how, then the factorization of $n$ is useless to find that of $n'$, and we are back to the problem of factorization of $n'$ without clue, which can be handled up to 800 bits (240 decimal digits) give or take 50%, depending on resources.