Great question. I move about every month and have this challenge, the key is planning. Unless in Asia or similarly cheap location I find accommodation with kitchen facilities. Then cook most of my meals by cooking big portions and reheating for a couple of days. I bring a couple of kitchen things like a thin cutting board, a knife, a container set, spices and some other things like rice, müsli, paper towel if I have left overs. Actually cook a great variety of excellent meals, just like before I starting traveling.
The biggest issue is not being allowed to bring liquid as carry-on on flights. I usually just bring a plastic bag with kitchen stuff on the plane. However not being able to bring liquid put some limitations on what I buy since I likely won't use it up and thus don't buy that, except if it's fairly cheap like soy sauce. However I've found that I just need to plan my meals more, so if I buy something I won't be able to bring I will make sure I make the dish twice in a big portion for it to make sense. This means a bit less variety, but more efficiency and still about 5-6 different dishes in a month and I just choose new ones the next month.
In a few days I'll move and there's only a few things I will throw out/leave behind, which I don't really mind. It takes a while to optimize and sometimes it doesn't go according to plan, but the consequences are pretty low. One example is planning a bit too much and ending up eating out more than anticipated in the end of a stay. I do safe a lot of money this way, have great meals all the time and it's pretty efficient.
Some essentials like cooking oil are best if they are already available in the accommodation, since it's rare to use a whole container in a month even if it's the smallest size available.
I had to teach myself how to cook a bunch of stuff in Hong Kong since I only had a pan and a microwave, but to my own surprise I managed to cook a bunch of different of my regular meals.
In cheap countries I always find accommodation with a fridge to be able to keep fruit, yoghurt and water.
One difference from having my own kitchen is the amount of spices, while I do keep the essentials (about 10 different ones) I don't have all those I need for all dishes. As a compromise I buy spice mixes/paste instead for a specific dish if I need it.