Or HaChaim on Genesis 24:4/אור החיים על בראשית כ״ד:ד׳

1 כי אל ארצי ואל מולדתי תלך, "rather you shall go to my country and to my birthplace, etc." This was part of the oath also; Eliezer was to secure a wife for Isaac in Abraham's birthplace. This is why Eliezer is described as immediately taking the camels, etc.