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Cause of Besuel's Death - Middah Kneged Middah
(נה) וַיֹּ֤אמֶר אָחִ֙יהָ֙ וְאִמָּ֔הּ תֵּשֵׁ֨ב הַנַּעֲרָ֥ אִתָּ֛נוּ יָמִ֖ים א֣וֹ עָשׂ֑וֹר אַחַ֖ר תֵּלֵֽךְ׃
(55) But her brother and her mother said, “Let the maiden remain with us some ten days; then you may go.”

(א) ויאמר אחיה ואמה. וּבְתוּאֵל הֵיכָן הָיָה? הָיָה רוֹצֶה לְעַכֵּב וּבָא מַלְאָךְ וֶהֱמִיתוֹ:

(1) ויאמר אחיה ואמה AND HER BROTHER AND HER MOTHER SAID —But where was Bethuel? He wished to prevent the marriage therefore an angel came and killed him (Genesis Rabbah 60:12).

(יב) וַיֹּאמֶר אָחִיהָ וְאִמָּהּ תֵּשֵׁב הַנַּעֲרָ אִתָּנוּ (בראשית כד, נה), וּבְתוּאֵל הֵיכָן הוּא, בִּקֵּשׁ לְעַכֵּב וְנִגַּף בַּלַּיְלָה, הֲדָא הוּא דִכְתִיב (משלי יא, ה): צִדְקַת תָּמִים תְּיַשֵּׁר דַּרְכּוֹ, צִדְקַת תָּמִים זֶה יִצְחָק, תְּיַשֵּׁר דַּרְכּוֹ שֶׁל אֱלִיעֶזֶר. (משלי יד, לב): וּבְּרָעָתוֹ יִדָחֶה רָשָׁע, זֶה בְּתוּאֵל שֶׁנִּגַּף בַּלַּיְלָה.

(א) ויאמר אחיה ואמה. פירש"י ובתואל רצה לעכב ובא מלאך והמיתו. ואע"ג דכתיב לעיל ויען לבן ובתואל צ"ל שרצה לחזור ולעכב הדבר.

(1) ויאמר אחיה ואמה, “Her brother and mother said;” Rashi, noticing that Betuel, Rivkah’s father is not quoted as making a response to Eliezer, says that he actually wanted to prevent the marriage so that an angel was dispatched by G–d to kill him in order to prevent him from carrying out his objection. He says so although the Torah had quoted both Lavan and Betuel as agreeing to the match being made in heaven in verses 50-51.

From Rashi's understanding of the Medrash, as well as how Daas Zekeinim understands Rashi, Besuel decided to renege on his decision to give Rivka over to Yitzchak in marriage and therefore he was smitten by an angel the night of his decision. The ways of Hashem are always "midda kneged mid", measure for measure, and the connection between his actions and the consequence he received seems to be because he wanted to stop Klal Yisrael's existence by preventing the destined Matriarch from going to Yitzchak and therefore he himself was cut off from his and his family's continuity.
However, some Rishonim as well as other Midrashim give a different spin on the story:

ד"א שבקשו להאכיל לאליעזר סם המות כדי שישאר להם הממון ובא גבריאל והחליף הקערה אשר בה סם המות ליתנה לאליעזר ונתנה לבתואל ומת:

An alternate interpretation explaining Betuel’s “silence.” He had meant to poison Eliezer so as to prevent the marriage. An angel switched plates while Eliezer was speaking and when Betuel started eating he died from his own poison.

(לג) וְסַדָרוּ קוּמֵיהּ לְמֵכוּל תַּבְשִׁילָה דְבֵיהּ סַמָא דִקְטוֹל וְאַרְגִישׁ בֵּיהּ וַאֲמַר לָא אֵיכוּל עַד דַאֲמַלֵיל פִּתְגָמָי וַאֲמַר מַלֵיל
(33) And he set in order before him to eat, prepared food in which was poison to kill; but he objected to it, and said, I will not eat, until I have spoken my words. And he said, Speak.

(נה) וְעַל דִהֲווֹ מְמַלְלִין בְּרַמְשָׁא בְּתוּאֵל הֲוָה אָכִיל מֵהַהוּא תַבְשִׁילָא וְאַשְׁכְּחוּנוֹהִי בִּקְרִיצְתָּא דְהָא מִית וַאֲמַר אֲחוּהָא וְאִימָא תִּיתַב רִיבָא עִמָנָא יוֹמֵי שַׁתָּא חֲדָא אוֹ עֲשַר יַרְחִין וּבָתַר כְּדֵין תֵּיזִיל

(55) But as they were talking in the evening, Bethuel had eaten of that prepared food; and in the morning they found that he was dead. And the brother and mother said therefore, Let the damsel dwell with us the days of one year or ten months, and then she shall go.
From these Rishonim, we see that an angel didn't directly smite Besuel, rather an angel "set up" Besuel to causers death by his own hands.
I think that this alternate explanation can even be the intention of the Medrash quoted by Rashi, because the Medrash says "וְנִגַּף בַּלַּיְלָה", "and he was smitten at night", which is parallel with the language the Targum Yonasan says in explaining the alternate explanation, "וְאַשְׁכְּחוּנוֹהִי בִּקְרִיצְתָּא דְהָא מִית", "and they found in the morning that he had died."
It implies that the way in which he died was by experiencing a reaction to the poison and in the morning it had completed its fatal effect.
But what is the middah kneged middah in this case? Why did Hashem use an angel to just facilitate Besuel to cause his own demise through the " poisonous plate switch" and not by the hands of the angel itself?
Perhaps we can say as follows: The Yefei Toar on the Medrash asks, we know that just moments before the Torah hints at Besuel's death because of his decision to renege, he and his son Lavan had just said:
(נ) וַיַּ֨עַן לָבָ֤ן וּבְתוּאֵל֙ וַיֹּ֣אמְר֔וּ מֵיְהֹוָ֖ה יָצָ֣א הַדָּבָ֑ר לֹ֥א נוּכַ֛ל דַּבֵּ֥ר אֵלֶ֖יךָ רַ֥ע אוֹ־טֽוֹב׃
(50) Then Laban and Bethuel answered, “The matter was decreed by יהוה; we cannot speak to you bad or good.
What had happened in a matter of minutes, or even seconds?
Yefei Toar says that in truth Besuel's retraction wasn't on the actual marriage of Rivka to Yitzchak. Rather, Besuel had a change of heart on the logistics. The plan was for Eliezer to bring Rivka to Yitzchak and Besuel changed his mind in this arrangement and demanded to have Yitzchak come in the flesh to pick up Rivka from her residence.
Perhaps it was for this reason Hashem caused Besuel to die in the way he did - measure for measure. Besuel wanted to switch the travel plans from Rivka going to Yitzchak to Yitzchak coming to Rivka. In the same way, Hashem caused the poisoned food he wanted to "send" to Eliezer, to "come to him" and cause his demise.