Chronic conditions and expat life coexist fine with some planning; millions of people manage diabetes, thyroid disorders, hypertension and more abroad. The groundwork happens before you fly: get a doctor's letter listing your diagnoses, medications by generic name and dosages, carry a good buffer of supply in original packaging, and split it between bags. Check your destination's rules for your specific medications, since insulin travels fine but some drugs are controlled substances in certain countries.
Once settled, register with a local doctor early rather than waiting for a crisis. In most countries a local physician can prescribe equivalents of your home medications after reviewing your records, and refilling locally is usually straightforward from then on, sometimes cheaper than at home. Brand names differ everywhere, so knowing generic names is essential, and formulations occasionally differ too, worth confirming with the pharmacist.
For diabetics specifically: pharmacies in most countries stock insulin, test strips and needles, though your exact pen or sensor brand may not be available, so research availability of your devices beforehand. Carry medical ID, keep insulin cool while traveling, and make sure your health insurance explicitly covers pre-existing conditions, because many travel policies exclude them by default.