I can't find in the documentation (manual, readme, Civilopedia) where the unit upgrade process is explained in detail. It is only implied in the Civilopedia entries for particular units (e.g. Horseman, Pikeman) and the Hotkeys are listed.
To upgrade (land) units from my experience:
o Unit to upgrade must be in a city with a barracks and supply of needed resources to build new unit, e.g. if upgrading a Spearman to a Pikeman, city must have barracks and be supplied with Iron.
o Select unit. Press U to upgrade. A dialog will appear prompting you to pay for upgrade.
-or-
o Select unit and press Shift+U to upgrade all units of that type (only units in cities that meet the criteria above will be affected). A dialog will appear prompting you to pay for upgrade.
To upgrade (land) units from my experience:
o Unit to upgrade must be in a city with a barracks and supply of needed resources to build new unit, e.g. if upgrading a Spearman to a Pikeman, city must have barracks and be supplied with Iron.
o Select unit. Press U to upgrade. A dialog will appear prompting you to pay for upgrade.
-or-
o Select unit and press Shift+U to upgrade all units of that type (only units in cities that meet the criteria above will be affected). A dialog will appear prompting you to pay for upgrade.
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