The two missing capitals are ...
... Muscat, the capital of Oman and Doha, the capital of Qatar.
The patterns ...
... consist either of eight colons and semi-colons or of eight upside-down semi-colons. There are not many patterns. They boil down to only six, sketched below. (I've used 9 and 6 here for the normal and upside-sown semicolons and a dot for the colon. That makes the patterns clearer.)
4 times (. . . . 9 . . .)
5 times (. . . 9 . . . .)
6 times (. . 9 . . 9 . .)
6 times (. 9 . . . . 9 .)
4 times (9 . . . . . . 9)
4 times (6 6 6 6 6 6 6 6)
Now if we ...
... organize the capitals by these patterns and also look up their countries, we get:
Astana (. . . . 9 . . .) Kazakhstan
New Delhi (. . . . 9 . . .) India
Niamey (. . . . 9 . . .) Niger
Athens (. . . . 9 . . .) Greece
? (. . . 9 . . . .)
Montevideo (. . . 9 . . . .) Uruguay
Quito (. . . 9 . . . .) Ecuador
San Salvador (. . . 9 . . . .) El Salvador
Belfast (. . . 9 . . . .) Northern Ireland
Brussels (. . 9 . . 9 . .) Belgium
Dublin (. . 9 . . 9 . .) Ireland
Khartoum (. . 9 . . 9 . .) Sudan
Budapest (. . 9 . . 9 . .) Hungary
? (. . 9 . . 9 . .)
Warsaw (. . 9 . . 9 . .) Poland
Bishkek (. 9 . . . . 9 .) Kyrgyzstan
Windhoek (. 9 . . . . 9 .) Namibia
Jakarta (. 9 . . . . 9 .) Indonesia
Tbilisi (. 9 . . . . 9 .) Georgia
Tegucigalpa (. 9 . . . . 9 .) Honduras
Port of Spain (. 9 . . . . 9 .) Trinidad & Tobago
Bucharest (9 . . . . . . 9) Romania
? (9 . . . . . . 9)
? (9 . . . . . . 9)
Nairobi (9 . . . . . . 9) Kenia
Lisbon (6 6 6 6 6 6 6 6) Portugal
Buenos Aires (6 6 6 6 6 6 6 6) Argentina
Cardiff (6 6 6 6 6 6 6 6) Wales
Oslo (6 6 6 6 6 6 6 6) Norway
We see ...
... that the patterns are the starting positions or files of the six chess pieces. The pawns (6) are on the second row or rank; the other pieces (9) are on the first row. (That's also what the title suggests: Capitals turned into pieces.)
The countries' first letters spell the names of the pieces, pawn, rook, knight, bishop, queen and king. Clearly, we are missing a Q for queen and and O for rook and bishop.
There is only one country that starts with O, Oman, and only one country that starts with Q, Qatar. The capitals of these two countries, Muscat and Doha, must fill the spaces.
