Compared with America, Britain is strangely deficient in leading Jewish poets. Elaine Feinstein is one of a very few, and her 12th collection is a unique project: a memoir through the lens of remembered cities, a lyric refraction of Judaism's uneasy nesting in Europe's cityscapes. She begins with Leicester, where she was brought up before the second world war, and Cambridge in the late 40s and 50s, where she was a student and a wife. Later she discovers the world's major cities – New York, Sydney, Dublin, Lisbon, Rome, Berlin – plus smaller towns. But the places that matter are London, where she now lives, and Warsaw, Krakow, Budapest, St Petersburg, Prague, Tbilisi and Odessa, where her grandparents were born. In a writing life spanning six decades, it is these Russia-haunted cities that call most from her now.

She encloses the urban, however, in nature. The first poem, "Migration", starts with "Mandelstam's goldfinch" and "birds from the Gambia, / white-throated warblers who wintered in / the branches of a feathery acacia", which migrate along "flyways old as Homer and Jeremiah". In the last poem, "Long Life", the poet, recovering from serious illness, contemplates "finches in the grass, and a stubby bush / which this year mothered a lemon".

Bird migration routes predate all human beings, not only Homer: cranes flew over the Trojan plain long before he compared their calls to those of an advancing army. But the birds are there to introduce the people. Just as an algebraic x placed before brackets enclosing an equation alters the value of everything within, "Migration" affects all the poems that follow. "Mandelstam's goldfinch" introduces one important theme, Russian poetry; the migrant "warblers" introduce another, the Greek and Jewish traditions that created western culture. The important "flyway" is the Jewish one, the line from Jeremiah to the Russian Jewish poet Osip Mandelstam, who once said that Judaism affected the Russian lyric tradition like a drop of musk filling the whole house. A poem called "Isaiah Berlin in Rome" is echoed by another, "Dizzy in Westminster", on the "baptised Jew" Disraeli. The vulnerably Judaic perspective of Feinstein's self-mapping-by-city reflects the city-rootedness of European Jews. Her journey recalls the 12th-century Spanish traveller Benjamin of Tudela, who gives details of Jewish communities in cities en route to the Holy Land, which are represented in Cities by Jerusalem in "the glow of a June sun, / until the desert night drops / a dark blue cloak over the streets".

For Feinstein, as for Benjamin, it is individuals who matter. Searching Odessa for traces of her grandfather, she finds Jewish "doctors and poets / TV journalists", who have returned to the burnt synagogues. Full of gestures and conversations, Cities reminds you what a city is – not only architecture or history but people.

This memoir-through-lyrics is the product of a working life that has frequently migrated between poetry and prose. Feinstein published her first poetry collection in 1966, her first novel (longlisted recently for the "lost" Man Booker prize) in 1970. She has also written six biographies and 14 novels which used to be called "poetic" – a word poets hate because they suspect it of meaning "woolly", but hers have been poetic in the sense of spare, wry, precise or tough. In assimilating lyric into prose like this, Feinstein found a literary correlative to Jewish assimilation in European cities: a way of staying yourself while becoming part of something else.

Feinstein has also migrated between eastern and western poetic influences. In the 50s, she explored new poetry from America and Russia. Signing herself in that pre-feminist world as "EB Feinstein", she wrote to the American poet Charles Olson, leader of the Black Mountain movement. In 1959 Olson wrote back. His "Letter to Mr Feinstein" elaborated his poetic principles to this enterprising and obviously male young poet. But the richest influence came from the east. In the 60s, the Russian poet Marina Tsvetayeva was untranslated. Feinstein liked the sound of her, collaborated with a Russianist, and published her acclaimed translations in 1971.

In Tsvetayeva, Feinstein found an inspiring entry into Russian poetry. The lyric gallantry which is a hallmark of Feinstein's work owes everything to Russia. Her poetry is itself a product of literary migration. In Mandelstam's Goldfinch, written in 1936 during his Voronezh exile, the little bird represents an unstoppable yearning for freedom. "I'll throw back my head," the poem says to the goldfinch. "Let's look at the world together." Feinstein's poems say something like this too. One of the most poignant is "Christmas Day in Willesden Green" (more assimilation to a Christian world). The poem, dedicated to her "autistic grandchild", tries to follow the boy's flyways, his "unknown pathways lying under speech". While rain brushes windows "like the crackle of crumpled cellophane", the 14-year-old cannot respond to his family at table. "His eyes are dark as wood resin/ . . . The scented candles and the roasted goose / with apples in its throat don't interest him. / He flicks a dangled string and sets it loose . . . / Is he bewildered / among so many strangers, or reconciled?"

These alternatives, reconciliation versus bewilderment among familial "strangers", underlie the brave humanity of this book, and its vision of a life lived through lyrics, through cities, and most deeply through and among other people.