I'm not a flaired user, so I've been reluctant to post - but this is a time period I'm intensely interested in and have done some research on.

My area of interest is Tokyo rather than Kyoto, but I've found some information relating to the move of the capital.

According to Edward Seidensticker, when talking about the ambiguities of the language (and the name Tokyo itself):

"A capital was 'established' in Edo or Tokyo. The capital was not, however 'moved' from Kyoto. So Kyoto, which means 'capital' could go on performing a role it was long accustomed to, that of vestigial or ceremonial capital. The Meiji emperor seems to have gone on thinking of Kyoto as his city; his grave lies within the Kyoto city limits."

It also seems that the move was gradual rather than sudden:

"The emperor departed Kyoto in the autumn of 1868. He reached the Shinagawa post station ...after a journey of some three weeks, and entered the castle on the morning of November 26.... The emperor returned to Kyoto early in 1869, after the pacification of the northern provinces. He was back in Tokyo in the spring, at which time his permanent residence may said to have begun. He did not announce that he was leaving Kyoto permanently, and the old capital went on expecting him back. It was not until 1871 that the last court offices were removed from Kyoto, and most of the court aristocracy settled in Tokyo."

Source: "Low City, High City" Tokyo from Edo to the Earthquake: how the shogun's ancient capital became a great modern city, 1867-1923. Edward Seidensticker 1983 Charles E. Tuttle Co. : Publishers

While this doesn't directly speak to the reactions of those in Kyoto, the ambiguity of the language as to which was the actual capital, the gradualness of the move and the fact that the Meiji emperor spent time there would seem to indicate that there wasn't much shock or upheaval and perhaps people though that things would essentially remain as they had been.