JS Bin// source https://jsbin.com/dowura/edit?js,output

jsbin.dowura.js

const {div, h, p, button, ul, li, makeDOMDriver} = CycleDOM;
const {Observable} = Rx;
const isolate = CycleIsolate;

function OriginalInput(sources) {
  const input$ = sources.DOM
    .events('input')
    .map(ev => ev.target.value);
  
  const vtree$ = sources.Assign
    .merge(input$)
    .startWith(null)
    .map(value =>
      h('input', {
        type: 'text',
        value
      })
    );
  
  return {
    DOM: vtree$,
    Values: input$
  };
}

const Input = s => isolate(OriginalInput)(s);

function main(sources) { 
  const add$ = sources.DOM.select('.add')
    .events('click').map(null);
  const input = Input({
    Assign: add$.map(() => ''),
    DOM: sources.DOM
  });

  const list$ = input.Values.flatMapLatest(text => add$.first().map(text))
    .startWith([])
    .scan((acc, curr) => acc.concat(curr));
  
  const vtree$ = list$.combineLatest(input.DOM, 
    (list, inputDOM) =>
      div([
        inputDOM,
        button('.add', 'Add'),
        ul(list.map(item => li(item)))
      ])
    );
  
  return {
    DOM: vtree$,
  };
}

Cycle.run(main, {
  DOM: makeDOMDriver('#app')
});

This is modification of A.Stalz's controlled input example (blog post , JSBin).

Instead of using virtual-dom hooks, I merge assign and DOM input sources into a value attribute stream.