Otdfbt

It is often said that Sri Chaitanya Mahaprabhu had predicted during his lifetime that His Sankirtana movement would spread throughout the entire world. In reality, the movement has indeed spread throughout the world thanks to the contributions of A.C Bhaktivedanta Swami Prabhupada and ISKCON. However, my question is that do any of the biographies of Sri Chaitanya Mahaprabhu indeed predict the Sankirtana movement to spreading throughout the entire world?

The most unambiguous reference comes in Antya Khanda of Chaitanya Bhagavata:

When Chaitanya Mahaprabhu arrived in Ramakeli news quickly spread everywhere, His loud roaring, kirtana, crying, and His appeal to everyone to chant the holy name of Hari
attracted the attention of even the infidels. When the local constable
related the topics of this wonderful sannyasi to the King, even that infidel
King Hussain Shah didn't want to disturb the sankirtana. Still, fearing that it was not impossible that the King's mind might change through the conspiracy of wicked people, some devotees secretly sent a messenger to the Chaitanya requesting Him to leave Ramakeli. When a brahmana informed the Lord's associates about this, their hearts filled
with anxiety. Then Chaitanya speaks many verses that there is no need to fear the Muslim king and he cannot do any harm. There Sri Chaitanya Mahaprabhu speaks:

prthivi-paryanta yata ache desa-grama sarvatra sancara haibeka mora nama

In every town and village throughout the world, the chanting of my
name will be spread. //CB Antya 4.126

Bhakti Siddhanta Sarasvati Thakur comments on the above verse as follows:

My names will be broadcast in all the villages and cities of the
world. Even though persons averse to the Supreme Lord have never heard
of the forms, qualities, and activities of the Supreme Lord, the holy
names of the Lord will be preached in all the villages of the world.

Until now, it has reached most of the countries and some towns, but not all the villages yet.

Other supportive statements can be, Chaitanya Charitamrta Madhya Lila

This statement occurs at the beginning of Sri Chaitanya Mahaprabhu's south Indian tour when kirtana was going on in Alalanatha.(a place near Jagannath Puri)

dekhi’ nityānanda prabhu kahe bhakta-gaṇe ei-rūpe nṛtya āge habe
grāme-grāme

Upon seeing the chanting and dancing of Lord Śrī Caitanya Mahāprabhu,
Lord Nityānanda said to the devotees that later there will be dancing and
chanting in every village. //CC Madhya 7.82

Another reference from Chaitanya Mangala, Sutra Khanda, Song 12((Ksudra-chanda - Dhanasi raga), this is shown as Krishna speaking to his associates before adverting as Chaitanya Mahaprabhu.

nama-guna-sankirtana vaisnavera shakti prakasa karibe ami nija
prema-bhakti

ei mate kali-papa karibe sanhara sabe cala age pache na kara vichara

ebe nama sankirtana tikshna khadaga laiya antara asura jivera phelibe
katiya

yadi papi chadi dharma dure dese yaya
mora senapati-bhakta yaibe tathaya

I will show them My holy names, My glories and transcendental
qualities, My sanirtana movement, and the power of My Vaisnava
devotees, I will show them the glory of the ecstatic spiritual love
the devotees feel for Me. In this way I will destroy the sins of
Kaliyuga. All of you please go there before Me. Go, and I will follow.
Don't hesitate. Wielding the sharp sword of nama-sankirtana (chanting
the holy names), I will cut to pieces the demonic desires in their
hearts. If the sinners giving up religious principles go to
far off places, the great devotees who are
generals in My armies will pursue them.

One such reference is found in the Chaitanya Bhagavata:

pṛthivīte āche yata nagarādi grāma
sarvatra pracāra haibe mora nāma

In every town and village throughout the world, the chanting of my name will be heard. (Chaitanya Bhagavata, Antya Khanda, 4.126)