The hidden 4-word phrase is:
A LETTER IN COMMON
The first thing I did was identify the countries and territories whose flags were depicted and also...
...list their 2-letter ISO codes (out of force of habit, I guess):
Bolivia (BO) + Isle of Man (IM)
Panama (PA) + Austria (AT)
Saint Helena (SH) + Guernsey (GG)
Canada (CA) + Ireland (IE)
Canada (CA) + Ireland (IE)
Saint Helena (SH) + Guernsey (GG)
Faroe Islands (FO) + Aruba (AW)
Slovakia (SK) + Cyprus (CY)
Burundi (BI) + Antigua & Barbuda (AG)
Peru (PE) + Argentina (AR)
Puerto Rico (PR) + Niger (NE)
Anguilla (AI) + Ethiopia (ET)
Anguilla (AI) + Ethiopia (ET)
Puerto Rico (PR) + Niger (NE)
Burundi (BI) + Antigua & Barbuda (AG)
Then I noticed that...
...there are 10 different flag pairs listed (some multiple times) - the same number as the number of what appear to be pairs of definitions in the list at the top of the puzzle. I reasoned that there may well be a one-to-one correspondence between the two, and so we should look for a way to connect the definition pairs and the flags.
I then spotted that...
...many of the definitions appeared to clue 3-letter words, e.g. 'came together' = MET, 'Companion' = PAL, 'Skilful and experienced' = PRO, 'single' = ONE...
Moreover, in each case, these words contained 2-letter sequences that appeared in the country ISO codes: M(ET), (PA)L, (PR)O, O(NE)...
And even more interestingly, within these words flags depicted on the left appeared at the left-hand end of a word - (PA)L, (PR)O - and flags depicted on the right appeared at the right-hand end of a word - M(ET), O(NE).
And still even more interestingly, for the two words that paired up - (PR)O and O(NE) - the flags also paired up, and the additional letter in both words was the same ('O').
This led me to a hypothesis:
Perhaps all of the definition pairs clue two 3-letter words of the form ABX and XCD, where 'AB' and 'CD' are the ISO codes of a pair of countries in the flag diagrams below, and 'X' is a common letter.
And it turned out I was right!
Note: Not all are strict definitions - some are more word association...
Skillful and experienced (PRO), single (ONE) = O
Flexible and agile (CAT), on a shirt (TIE) = T
"To" (FOR), in it's natural state (RAW) = R
Chest (PEC), vehicle (CAR) = C
Shoot (AIM), came together (MET) = M
Companion (PAL), part of the back (LAT) = L
Rubbish (BIN), annoy (NAG) = N
Miss is a? (SHE), a future animal inside (EGG) = E
Long animal (BOA), shoot (AIM) = A
On snow (SKI), cold (ICY) = I
Finally, we can...
...decode the flag pairs by replacing them with the 'common letter' that they map to in the corresponding clues:
Bolivia (BO) - A - Isle of Man (IM)
Panama (PA) - L - Austria (AT)
Saint Helena (SH) - E - Guernsey (GG)
Canada (CA) - T - Ireland (IE)
Canada (CA) - T - Ireland (IE)
Saint Helena (SH) - E - Guernsey (GG)
Faroe Islands (FO) - R - Aruba (AW)
Slovakia (SK) - I - Cyprus (CY)
Burundi (BI) - N - Antigua & Barbuda (AG)
Peru (PE) - C - Argentina (AR)
Puerto Rico (PR) - O - Niger (NE)
Anguilla (AI) - M - Ethiopia (ET)
Anguilla (AI) - M - Ethiopia (ET)
Puerto Rico (PR) - O - Niger (NE)
Burundi (BI) - N - Antigua & Barbuda (AG)
Which read in order spells out the answer:
A LETTER IN COMMON - which is exactly what these 3-letter country-substring words have!