Erasure codes have become an integral part of distributed storage systems as a tool for providing data reliability and durability under the constant threat of device failures. In such systems, an $[n, k]$ code over a finite field $\mathbb{F}_q$ encodes $k$ message symbols into $n$ codeword symbols from $\mathbb{F}_q$ which are then stored on $n$ different nodes in the system. Recent work has shown that significant savings in storage space can be obtained by tuning $n$ and $k$ to variations in device failure rates. Such a tuning necessitates code conversion: the process of converting already encoded data under an initial $[n^I, k^I]$ code to its equivalent under a final $[n^F, k^F]$ code. The default approach to conversion is to reencode data, which places significant burden on system resources. Convertible codes are a recently proposed class of codes for enabling resource-efficient conversions. Existing work on convertible codes has focused on minimizing access cost, i.e., the number of code symbols accessed during conversion. Bandwidth, which corresponds to the amount of data read and transferred, is another important resource to optimize. In this paper, we initiate the study on the fundamental limits on bandwidth used during code conversion and present constructions for bandwidth-optimal convertible codes. First, we model the code conversion problem using network information flow graphs with variable capacity edges. Second, focusing on MDS codes and an important parameter regime called the merge regime, we derive tight lower bounds on the bandwidth cost of conversion. The derived bounds show that bandwidth cost can be significantly reduced even in regimes where access cost cannot be reduced as compared to the default approach. Third, we present a new construction for MDS convertible codes which matches the proposed lower bound and is thus bandwidth-optimal during conversion.