Jet Set

The first thing to notice is that each postcard describes a three-letter abbreviation in medicine. The first card, for example, reads:

"A man told me, while we waited in the transfer lounge, about how he once went into a large donut-looking machine, to have this scan which used arterial contrast to locate clots or aneurysms."

This clues for CT Angiogram, or CTA for short.

It turns out that airports also have three-letter abbreviations: the IATA code (or, if that Wiki link is too boring, here's a CGP Grey video about them instead). For the above card, CTA is the IATA code for Fontanarossa Airport in Catania, Italy. This explains the Italian flag on the postcard's stamp; the presence of these flags help us confirm we have the right three-letter abbreviations for the cards.

To order the postcards to reconstruct the itinerary, note the presence of two cards that clearly identify the start and end of the trip. Respectively:

"So excited to start my world tour from here! But the airport's so small, I had nothing to do except revise the pharmacological approach to myocardial infarcts, of which this drug is a vasodilator." (GTN)

"Here is my last stop before going home. Thankfully I remember the corrected name of the disease I most recently studied; just in time for the CAT!" (JIA)

The trick with ordering is that adjacent postcards differ by only one letter in the three-letter abbreviations. There is exactly one way of ordering these postcards, including the five damaged ones, under this rule.

Clue	Abbreviation
So excited to start my world tour from here! But the airport's so small, I had nothing to do except revise the pharmacological approach to myocardial infarcts, of which this drug is a vasodilator.	GTN
Walking off the plane made me appreciate these proprioceptors located at my musculotendinous junctions, without which I can't walk!	GTO
????	GTA / STO
Had to run to make this flight! As I sat down I could really feel pulsations in this terminal branch of my external carotid.	STA
A man told me, while we waited in the transfer lounge, about how he once went into a large donut-looking machine, to have this scan which used arterial contrast to locate clots or aneurysms.	CTA
I'm glad Dermacentor andersoni is not endemic in Central America; I don't want to contract this disease that it transmits!	CTF
Stumped on a crossword clue from the newspaper I got from the terminal newagent: "This protein transport iron to cells, and can be found in milk"	LTF
Ouch! My diverticulitis must be playing up again, because I'm feeling pain in this specific region of my abdomen, if you divide my abdomen into nine zones.	LIF
????	LCF / LIM
I just saw a house mouse scurry across the terminal! I hope it doesn't carry any Arenavirus that could give me this infection!	LCM
Trying to memorise all subtypes of this disease, of which "Hand–Schüller–Christian" is one. Gosh I hate eponymous naming. You'd think they were trying to name an airport!	LCH
Had to run to catch my flight again from a tiny island airport. Can't help but think about how hard this enzyme, that catalyses pyruvate in anaerobic reactions, is working right now.	LDH
There was a crying baby the entire flight! Just glad for them that they didn't have this paediatric disease where abdominal organs would push into the chest cavity because of a defect.	CDH
????	CDF / MDH
Air travel really detracts from efforts to fulfil number 7 on this list of objectives set by the United Nations in 2000.	MDG
Thankfully the terminal had hand sanny dotted around. Don't wanna bring home any MRSA, VRE, ESBLs etc, which have this property where many antimicrobials don't work.	MDR
Flipping through my stats textbook because I hate myself, I can't help but notice this term for Type I error shares an abbreviation with the WWII President of the country I'm currently in.	FDR
????	FLR / PDR
Opened the window of the plane and was hit by sunlight. I hope the diameters of my eye apertures appropriately changed as a result of this mechanism.	PLR
Visiting an area of the tropics, I am thankful for this germline-encoded host sensing molecule, which detect typical pathogens I might encounter.	PRR
The parents in front of me at Security were carrying a baby with this micrognathia, glossoptosis and airway obstruction.	PRS
????	JRS / PRA
Today I read about this childhood condition that causes joint pains, but I think the name I recorded in my Anki is outdated... unfortunately the helipad had no WiFi for me to check.	JRA
Here is my last stop before going home. Thankfully I remember the corrected name of the disease I most recently studied; just in time for the CAT!	JIA

There is some ambiguity with which three-letter abbreviation to pick for the damaged cards (eg. going from GTO to STA can pass through either STO or GTA). Let's try and see what our options are:

Damaged card	Option 1	Option 2
#1	GTA - Gatokae	STO - Stockholm
#2	LCF - Río Dulce	LIM - Lima
#3	CDG - Paris	MDH - Carbondale/Murphysboro
#4	FLR - Florence	PDR - Presidente Dutra
#5	JRS - Jerusalem	PRA - Parana

In each case, one option is a major city, while the other is a small remote airport. So assume we use the major cities.

The flavour text asks about what links these places together. And indeed, they are each an example of a SYNDROME named after a place. For example, Stockholm Syndrome is an alleged situation where a kidnapping victim grows to sympathise with their captor.