It is a good idea to keep the same basics of your brand. However, mixing it with the local culture a little bit is always a good idea. For example, Starbucks is known for its amazing range of beverages, especially coffee. However, they were not a success in China as people prefer to drink tea there. Hence, the brand tweaked their menu a little bit and started offering tea as well. This is how it was possible for them to sustain in China.
Thus, it is important to have a localisation strategy. In fact, the food industry is highly based on localisation strategy in most parts of the world. This is because the food habits vary from region to region. However, if you are not a food brand, you can still imply this strategy and benefit from it. It is one of the best ways to perform global branding.