We are looking for 4 countries/kingdoms/sovereign states which do not exist anymore.
ISO-3166-1 alpha-2 codes are of use here
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Once the OP's comment got me away from thinking about dates, this just took a lot of effort, not too much thinking. Thanks to the hint, we have the following values for the flags:
Chile = CL, Bolivia = BO, Netherlands Antilles = AN, USVI = VI
UN = UN, Germany = DE, Austria = AT, Slovenia = SI
Boznia-Herzegovina = BA, Eritrea = ER, Ethiopia = ET, Armenia = AM
Armenia = AM, Somalia = SO, Yemen = YE, BIOT = IO, Nepal = NP
In the original order, that's
CL BO AN VI UN DE AT SI BA ER ET AM AM SO YE IO NP
Putting those in the correct order for the question, we get
CL + AN + DE + BO + YE + SO + VI + ET + UN + IO + NP + ER + SI + AM + AT + AM + BA
which results in
Let's get the ball rolling with a partial answer... Here are the 17 flags identified:
And here are some initial observations:
1. Reading along each row, the flags appear to follow a west-east route across the globe (ignoring Armenia's double appearance, and basing the United Nations at its headquarters in New York, USA).
2. Many of the flags can also be grouped geographically - loosely they fall into the following geographical categories:
Horn of Africa: Ethiopia, Eritrea, Somalia;
South America: Chile, Bolivia;
Europe: Germany, Austria, Slovenia, Bosnia-Herzegovina;
Asia: Armenia x2, Yemen, Nepal;
Island groups: Netherlands Antilles, US Virgin Islands, British Indian Ocean Territory;
Organisations: United Nations
3. It might also be possible to try to draw connections purely with the countries represented in each row - for example, drawing a line connecting the four in the first row on a map ends up dividing South America and the Caribbean in an almost north-south fashion, while it is also possible to draw a line on a map through the second, third and fourth countries in the second row without crossing any other countries' borders (all 3 were also part of the Habsburg family empire, historically speaking).
It is unclear to me at this stage whether the OP has selected the countries in the list above for geographical reasons (e.g. to identify the land area occupied by a particular ex-kingdom on a map) or because their names satisfy another property and choosing countries which border each other is a good way to sow a false trail... In particular:
The inclusion of the flag of the United Nations (not a geographical entity) makes me think the puzzle solution will not depend on geography, but potentially either history (e.g. year of independence/formation) or wordplay (e.g. using certain letters in each name).
Also, the fact that the flag of Armenia is deliberately utilised twice makes me lean towards something to do with the names of the entities these flags represent (e.g. initials), potentially on a per-row basis. However, quite why Armenia is used twice rather than, say, using Albania (a very similar name, spelling-wise) for one of them is currently beyond me - I'm sure the link will become clear soon...
That's the scutwork done - where to now??
EDIT: Following the OP's hint...
The ISO-3166-1 alpha-2 codes for each of the entities represented in these flags are:
CH / BO / AN / VI /
UN / DE / AT / SI /
BA / ER / ET / AM /
AM / SO / YE / IO / NP
I note that it is possible to concatenate many of these into answers which may satisfy this puzzle's brief (ex-countries, former kingdoms, etc.). For example, SI+AM for Siam, the former name of Thailand. A longer string might be BA+DE+NP+ER+SI+AN+AM+VI+ET, yielding Baden (a former European state), Persia (the ancient kingdom predating modern-day Iran) and Nam Viet (an ancient kingdom also called Nanyue, which covered parts of modern-day China and Vietnam). However, this still leaves a lot of letter pairs unaccounted for - rather too many for the one remaining answer, in all likelihood... I shall think some more.
I might have a thing here.
In the second row there are the following Flags with the dates of foundation / independency
UN: 1945 (Foundation)
Germany: 1949 (Federal Republic)
Austria: 1945 (Second Republic)
Slovenia: 1991 (Independence)
The arithmetic mean of those four dates is $$(1945 + 1949 + 1945 + 1991) / 4 = 1957.5$$. In the year 1975 the Saarland got re-connected with Germany
So here are the data for the whole bunch of Flags
Note: I used a standardized search on Google which gave me some different numbers
Chile: September 18, 1810
Bolivia: August 6, 1825
Netherlands Antilles: 1954
US Virgin Islands: March 31, 1917
United Nations: October 24, 1945
Germany: October 3, 1990
Austria: July 27, 1955
Slovenia: October 29, 1918
Bosnia Herzegovina: March 1, 1992
Eritrea: April 27, 1993
Ethiopia: August 21, 1995
Armenia: September 21, 1991
Armenia: September 21, 1991
Somalia: 2500 BC
Yemen: October 30, 1918
British Indian Ocean Territory: 1965
Nepal: September 25, 1768
Here's the number crunching part
$1810 + 1825 + 1954 + 1917 = 7506$
$7506 / 4 = 1876,5$
$1945 + 1990 + 1955 + 1918 = 7808$
$7808 / 4 = 1952$
$1992 + 1993 + 1995 + 1991 = 7971$
$7971 / 4 = 1992,75$
$1991 - 2500 + 1918 + 1965 + 1768 = 5142$
$5142 / 5 = 1028,4$
I don't know. if this is pointing in the right direction
Maybe it could be related to the UN itself, but the US Virgin Islands and the British Indian Ocean Territory and the Netherlands Antilles are not members of th UN