We present a detailed characterization of the mineralogy, petrology, and oxygen isotopic compositions of twelve FUN CAIs, including C1 and EK1-4-1 from Allende (CV), that were previously shown to have large isotopic fractionation patterns for magnesium and oxygen, and large isotopic anomalies of several elements. The other samples show more modest patterns of isotopic fractionation and have smaller but significant isotopic anomalies. All FUN CAIs studied are coarse-grained igneous inclusions: Type B, forsterite-bearing Type B, compact Type A, and hibonite-rich. Some inclusions consist of two mineralogically distinct lithologies, forsterite-rich and forsterite-free/poor. All the CV FUN CAIs experienced postcrystallization open-system iron-alkali-halogen metasomatic alteration resulting in the formation of secondary minerals commonly observed in non-FUN CAIs from CV chondrites. The CR FUN CAI GG#3 shows no evidence for alteration. In all samples, clear evidence of oxygen isotopic fractionation was found. Most samples were initially 16O-rich. On a three-oxygen isotope diagram, various minerals in each FUN CAI (spinel, forsterite, hibonite, dmisteinbergite, most fassaite grains, and melilite (only in GG#3)), define mass-dependent fractionation lines with a similar slope of ∼0.5. The different inclusions have different Δ17O values ranging from ∼−25‰ to ∼−16‰. Melilite and plagioclase in the CV FUN CAIs have 16O-poor compositions (Δ17O ∼−3‰) and plot near the intercept of the Allende CAI line and the terrestrial fractionation line. We infer that mass-dependent fractionation effects of oxygen isotopes in FUN CAI minerals are due to evaporation during melt crystallization. Differences in Δ17O values of mass-dependent fractionation lines defined by minerals in individual FUN CAIs are inferred to reflect differences in Δ17O values of their precursors. Differences in δ18O values of minerals defining the mass-dependent fractionation lines in several FUN CAIs are consistent with their inferred crystallization sequence, suggesting these minerals crystallized during melt evaporation. In other FUN CAIs, no clear correlation between δ18O values of individual minerals and their inferred crystallization sequence is observed, possibly indicating gas-melt back reaction and oxygen-isotope exchange in a 16O-rich gaseous reservoir. After oxygen-isotope fractionation, some FUN CAIs could have experienced partial melting and gas-melt oxygen-isotope exchange in a 16O-poor gaseous reservoir that resulted in crystallization of 16O-depleted fassaite, melilite and plagioclase. The final oxygen isotopic compositions of melilite and plagioclase in the CV FUN CAIs may have been established on the CV parent asteroid as a result of isotope exchange with a 16O-poor fluid during hydrothermal alteration.We conclude that FUN CAIs are part of a general family of refractory inclusions showing various degrees of fractionation effects due to evaporative processes superimposed on sampling of isotopically heterogeneous material. These processes have been experienced both by FUN and non-FUN igneous CAIs. Generally, the inclusions identified as FUN show larger isotope fractionation effects than non-FUN CAIs. There is a wide spread in UN isotopic anomalies in a large number of CAIs not exhibiting large fractionation effects in oxygen, magnesium, and silicon. The question of why some FUN CAIs show more extreme UN isotopic effects is attributed by us to limited sampling and not a special source of isotopically anomalous material. We consider the majority of igneous CAIs to be the result of several stages of thermal processing (evaporation, condensation, and melting) of aggregates of solid precursors composed of incompletely isotopically homogenized materials. The unknown nuclear effects in CAIs are common to both FUN and non-FUN CAIs, and are not a special characteristic of FUN inclusions but represent the spectrum of results from sampling a very heterogeneous medium in the accreting Solar System.