So there are two main episodes that the Eiruv Rav (mixed multitude) were instrumental in:
1) Sin of the Golden Calf
In Shemos 32:4 we are told:
וַיִּקַּ֣ח מִיָּדָ֗ם וַיָּ֤צַר אֹתוֹ֙ בַּחֶ֔רֶט וַֽיַּעֲשֵׂ֖הוּ עֵ֣גֶל מַסֵּכָ֑ה וַיֹּ֣אמְר֔וּ אֵ֤לֶּה אֱלֹהֶ֙יךָ֙ יִשְׂרָאֵ֔ל אֲשֶׁ֥ר הֶעֱל֖וּךָ מֵאֶ֥רֶץ מִצְרָֽיִם׃
This he took from them and cast in a mold, and made it into a molten calf. And they exclaimed, “This is your god, O Israel, who brought you out of the land of Egypt!”
אלה אלהיך THESE ARE THY GODS, [O ISRAEL] — It does not state that they said “these are our gods, [O Israel]”; hence we may learn that it was the mixed multitude which came up from Egypt that gathered themselves together against Aaron, and it was they who made it and afterwards led Israel astray after it (Midrash Tanchuma, Ki Tisa 19).
This Midrash as well as the Chizkuni note that they didn't just force through the construction of the golden calf but the sorcerers amongst them made it look that it was communicating back to the people thereby making it all the more believable that it was a suitable replacement for Moshe.
2) They were the מתאוננים - The Complainers
In Bamidbar 11:4 we're told:
וְהָֽאסַפְסֻף֙ אֲשֶׁ֣ר בְּקִרְבּ֔וֹ הִתְאַוּ֖וּ תַּאֲוָ֑ה וַיָּשֻׁ֣בוּ וַיִּבְכּ֗וּ גַּ֚ם בְּנֵ֣י יִשְׂרָאֵ֔ל וַיֹּ֣אמְר֔וּ מִ֥י יַאֲכִלֵ֖נוּ בָּשָֽׂר׃
And the gathering that was among them fell a lusting; and the children of Israel also wept on their part, and said: ‘Would that we were given flesh to eat!
Rashi qualifies that this gathering was the Eiruv Rav:
והאספסף AND THE THRONG — This was the mixed multitude that had gathered themselves unto them when they left Egypt (the word is from the root אסף, “to gather”) (Sifrei Bamidbar 86)...
Their complaining then triggered a domino effect on the rest of the Jewish people. Chizkuni asserts:
וישובו ויבכו גם בני ישראל, “and in due course even the elite of the people, the בני ישראל, also joined the weeping of discontent that had been started by the mixed multitude. This is when they expressed their urge to be given meat to eat.
Furthermore, they weren't just asking and complaining for meat. According to Sforno there was a more deep-seated desire:
מי יאכלנו בשר?, They said this in order to test if He would permit incestuous relations to His people (compare Psalms 78:18 where Assaph says וינסו א-ל בלבבם לשאל אוכל לנפשם, they asked openly for meat, and in their hearts for sexual gratification other than by their wives).