I just inverted what you said without changing the meaning. If you say X is greater than Y and I reply that you'd said that Y was less than X doesn't change the meaning.
That said, a larger alphabet (adding more possible characters) increases the number of possible combinations (what I think you're calling a 'search set').
From a combinatorics viewpoint, the lower case alphabet alone gets you 26 symbols, so there are 1.56 E+6 possible 8-char subsets. Add in upper case, numbers, and a few symbols to get to 72 symbols and that goes up to 1.196 E+10, so about 4 orders of magnitude more possible subsets.
BTW, using lower case but increasing password length to 10 increases the number of subsets from 1.56 to 5.31 (both E+6).
On to XKCD's passphrase correcthorsebatterystaple. A quick lookup shows there are about 1000 common words in English (which tracks with the number of word glyphs you'd need to recognize to read a newspaper in Mandarin). So an alphabet of 1000 x 4 'letters' gets us 4.14 E+10. Maybe double that for all lower case vs CamelCase - but the O is what matters.
BTW, I'm using simple discrete math for this, so set selection, not taking into account that characters might repeat - or all the different ways you can arrange those 8 (or 4) 'letter' subsets. IMO suitable for back-of-the-envelope estimates.