Let me start with the source I am most familiar with, namely, Tanya, from Kavod Kedushas ADMO”R HaZoken ZY”A.
In Perek 2 it states: “for the nuture and life-force of the nefesh, ruach, and neshamah of the amei haaretz are drawn from the nefesh, ruach, and neshamah of the righteous and the sages, the “heads” of Israel in their generation.” “This explains the comment of our Sages on the verse, “And cleave unto Him (which the sages explain in ketuvos 111b means, “He who cleaves unto a Torah scholar is deemed by the Torah to have mamesh become attached to the Shechinah”
Thus by following this reasoning, one could reason that if tzadikim are indeed where the “nurture” and “life-force” of the general Jewish population are drawn from, and that how the Jewish people behave effects how Hashem chooses to interact with the world (as I brought b’loshon Nefesh HaChaim in an earlier post), then one can see that tzadikim directing the Jewish people and connecting them to Hashem effects how Hashem chooses to run the world. Which is what I was claiming above, and I believe how Kavod Kedushas ADMO”R of Toldos Avrohom Yitzchok interpreted the quote from his Rebbe ZY”A. Thus, it would follow that Rebbeim do indeed have an effect on this world, which was the entire intention of my original post to come and prove.
Furthermore, and this is more shakey as I do not have the sefer in front of me, nor do I have access to the sefer at this moment: but as I recall, it quotes in Kedushas Noam Elimelech (not to be confused with stam Noam Elimelech. This sefer has an added section of Rebbeim from the next generation giving praises about the Baal Noam Elimelech ZY”A.), from Kavod Kedushas ADMO”R from Belz ZY”A (could be wrong in this attribution of the quote, but the quote exists there) stating, “all the teshuva which is done in this world is done with the strength of the Rebbe Rebbe Elimelech, for it has not passed one hundred years since his passing.” V’Chu V’Chu.
Furthermore, there’s simply this quote from Moed Katan 16b:
אמר אלהי ישראל לי דבר צור ישראל מושל באדם צדיק מושל יראת אלהים מאי קאמר א”ר אבהו ה”ק אמר אלהי ישראל לי דבר צור ישראל אני מושל באדם מי מושל בי צדיק שאני גוזר גזרה ומבטלה