To become extremely hot and sweaty; to swelter or cause to swelter thoroughly.
From prefix 'be-' + 'swelter' (possibly from Middle Dutch 'swelten,' to melt or faint). The 'be-' intensifies the verb to mean 'to cause extreme heat.'
Imagine Victorian writers describing someone 'besweltering' in heavy clothing on a summer day — it conveys not just being hot, but being overwhelmingly, almost unbearably hot in a very descriptive way.
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