The link between the cognitive effort of word processing and the eye-movement patterns elicited by that word is well established in psycholinguistic research using eye-tracking. Yet less evidence or consensus exists regarding whether the same link exists between linguistic complexity measures of a sentence or passage and eye movements registered at the sentence or passage level. This article focuses on "global" measures of syntactic and lexical complexity, i.e., the measures that characterise the structure of the sentence or passage rather than aggregate lexical properties of individual words. We selected several commonly used global complexity measures and tested their predictive power against sentence- and passage-level eye movements in samples of text reading from 13 languages represented in the Multilingual Eye Movement Corpus (MECO). While some syntactic or lexical complexity measures elicited statistically significant effects, they were negligibly small and not of practical relevance for predicting the processing effort either in individual languages or across languages. These findings suggest that the "eye-mind" link known to be valid at the word level may not scale up to larger linguistic units.