Why did Moshe Rabbeinu daven for Yehoshua, and leave Kaleiv on his own to fend for himself? The Chofetz Chaim (on the Torah) explains that Moshe realized that Yehoshua and Kaleiv had opposite approaches. Yehoshua's approach was to fight for what he beleived in, and not even for a second give the impression that he agrees with the meraglim. Kaleiv's approach was logistical tactics that would be most effective in winning the litigation, which involved giving the impression that he agrees until the ideal opportunity presents itself to express his true feelings. Moshe Rabbeinu was concerned that with Yehoshua's antagonistic method, his life was endangered because the mergalim may try to harm him, therefore he davened fro him. Kaleiv on the other hand pretended to be on their side until they returned. As Rashi writes on the pasuk - ועבדי כלב עקב היתה רוח אחרת עמו, his heart did not match his mouth; he pretended to agree with the meraglim until the opportune time to quiet them down. However, when they began conceiving their hideous plans, Kaleiv was concerned they by pretending to agree with them he may actually become influenced and felt it was necessary to protect himself by going to chevron to beseech the avos to protect his convictions.
I would like to add to this approach and explain why each one chose the tactic that they did. The major tactic that Kaleiv used is to pretend that he was also against Moshe, as we see in the gemara 35a - אמר להן, וכי זו בלבד עשה לנו בן עמרם, סברי בגנותיה קא משתעי, אשתיקו, אמר להו הוציאנו ממצרים וקרע לנו את הים והאכילנו את המן וכו
Yehoshua was unable to stomach such a tactic because Yehoshua could not tolerate any denigration of Moshe as we see when eldad and meidad were saying nevuah -
ויען יהושע בן נון משרת משה מבחריו ויאמר, "אדני משה כלאם"- פרק יא פסוק כח
The Meshech Chochma explains that Kaleiv's not only used this as a tactic but truly believed that the success of klal yisroel is mutually exclusive from Moshe Rabbeinu. The Pasuk says:
ויהס כלב את העם אל משה, ויאמר עלה נעלה וירשנו אותה כי יכול נוכל לה
What does it mean - "אל משה"? It means that he was downplaying Moshe's role. Part of the meraglim's claim is that without Moshe leading them into Eretz Yisroel they won't be able to conquer it, to which kaleiv responds, "We don't need moshe, we can go up and succeed with or without Moshe". Basically, Kaleiv's whole approach was based on his ability to give the impression that he was against Moshe, and actually downplaying the need for Moshe. Yehoshua, as an absolute talmud muvhak of Moshe could not take any approach that would degrade Moshe at all, and was therefore forced to use the antagonistic tactic of fighting with the meraglim directly. Moshe realized this and therefore only davened for Yehoshua and not Kaleiv.