How do I push several to-dos at once, and why is "Push" sometimes missing from the bar?

Select them and click "Push" — they're sent one after another to your connected apps (respecting each list's chosen apps), with a progress message. Anything that can't be pushed is skipped (items still saving, already pushed, or in a locked list). Push only appears when your selection is entirely free of locked-list items and contains at least one pushable to-do — if even one selected to-do is in a padlocked list, Push hides completely. Remove it and Push reappears.