To achieve this please specify base.

¶ base - Specify the folder relative to the cwd. Default is where the glob begins. This is used to determine the file names when saving in .dest()

In your case it would be:

gulp.task('move-css',function(){
  return gulp.src([
      './source/css/one.css',
      './source/other/css/two.css'
  ],  {base: './source/'}) 
  .pipe(gulp.dest('./public/assets/'));
});

Folder structure:

.
├── gulpfile.js
├── source
│   ├── css
│   └── other
│       └── css
└── public
    └── assets

I use gulp-flatten and use this configuration:

var gulp = require('gulp'),
    gulpFlatten = require('gulp-flatten');

var routeSources = {
  dist:  './public/',
  app: './app/',
  html_views: {
    path: 'app/views/**/*.*',
    dist: 'public/views/'
  }
};

gulp.task('copy-html-views', task_Copy_html_views);
function task_Copy_html_views() {
  return gulp.src([routeSources.html_views.path])
      .pipe(gulpFlatten({ includeParents: 1 }))
      .pipe(gulp.dest(routeSources.html_views.dist));
}

And there you can see the documentation about gulp-flatten: Link

   gulp.task('move-css',function(){
      return gulp
                  .src([ 'source/**'], { base: './' })
                  .pipe(gulp.dest('./public/assets/css/'));
    });

Your own code didn't include the entire dir tree of source 'source/**' and the base {base:'./'} when calling to gulp.src which caused the function to fail. The other parts where fine.

gulp.task('move-css',function(){
  return gulp.src([
    './source/css/one.css',
    './source/other/css/two.css'

    ]).pipe(gulp.dest('./public/assets/css/'));
});