Alliteration is used sporadically throughout "The Emigrée." It is first used in line 1:
There once was a country… I left it as a child
The /w/ sound adds to the sing-song, fairy-tale like opening statement of the poem. Lines 2 and 3 then alliterate on the /m/, /n/, and /s/ sounds:
but my memory of it is sunlight-clear
for it seems I never saw it in that November
This cluster of alliteration continues the poem's lyrical tone. The opening lines sound pretty and poetic—which makes sense, given that the speaker is describing her beloved home.
The next significant alliteration is in line 8, with: "but I am branded by an impression of sunlight." This relates to the way that the speaker experiences her memory: it is something permanent and unshakeable, a source of strength. For this reason, it is "branded" on her life. To match, the line is itself branded by plosive /b/ sounds. These sounds, however, in their connection to the word "branded," also hint at torture and violence (which is one of the most common reasons why people have to emigrate in the first place).
As if in reference to line 8's alliteration, lines 15 and 16 return to this /b/ sound. Here, the speaker is describing forms of oppression used by the state to exert its power. The /b/ sounds dominate the line, barely allowing other sounds to exist:
It may by now be a lie, banned by the state
but I can’t ...
The alliterating /t/ sounds in "tongue" and "tastes" come along like a kind of relief from this /b/ sound, and emphasize the intensity of the speaker's memories.