Adding to @Jonas and @KFunk's answer:
Potential fix for requiring the use of all col-sizes (col-xs-6 col-sm-4 col-md-4 col-lg-4).
Classes created: auto-clear-xs, auto-clear-sm, auto-clear-md and auto-clear-lg. 
I've made my answer mobile-first.
Note: This still requires the columns to be the same size.
HTML
<div class="row auto-clear-xs auto-clear-lg">
    <div class="col-xs-6 col-lg-3">
        <p>Hey</p>
    </div>
</div>
SCSS
@mixin row-first-child($col-type, $clear-type) {
   .col-#{$col-type}- {
    &1:nth-child(12n+1),
    &2:nth-child(6n+1),
    &3:nth-child(4n+1),
    &4:nth-child(3n+1),
    &6:nth-child(odd){
      clear: $clear-type;
    }
  }
}
.auto-clear-xs{
  @media (min-width: $screen-xs-min){
    @include row-first-child(xs, both);
}
  @media (max-width: $screen-xs-min){
    @include row-first-child(xs, both);
  }
}
.auto-clear-sm{
  @media (min-width: $screen-sm){
    @include row-first-child(xs, none);
    @include row-first-child(sm, both);
  }
}
.auto-clear-md{
  @media (min-width: $screen-md){
    @include row-first-child(xs, none);
    @include row-first-child(sm, none);
    @include row-first-child(md, both);
  }
}
.auto-clear-lg{
  @media (min-width: $screen-lg){
    @include row-first-child(xs, none);
    @include row-first-child(sm, none);
    @include row-first-child(md, none);
    @include row-first-child(lg, both);
  }
}
CSS
@media (min-width: 480px) {
  .auto-clear-xs .col-xs-1:nth-child(12n+1), 
  .auto-clear-xs .col-xs-2:nth-child(6n+1), 
  .auto-clear-xs .col-xs-3:nth-child(4n+1), 
  .auto-clear-xs .col-xs-4:nth-child(3n+1), 
  .auto-clear-xs .col-xs-6:nth-child(odd) {
    clear: both;
  }
}
@media (max-width: 480px) {
  .auto-clear-xs .col-xs-1:nth-child(12n+1), 
  .auto-clear-xs .col-xs-2:nth-child(6n+1), 
  .auto-clear-xs .col-xs-3:nth-child(4n+1), 
  .auto-clear-xs .col-xs-4:nth-child(3n+1), 
  .auto-clear-xs .col-xs-6:nth-child(odd) {
    clear: both;
  }
}
@media (min-width: 768px) {
  .auto-clear-sm .col-xs-1:nth-child(12n+1), 
  .auto-clear-sm .col-xs-2:nth-child(6n+1), 
  .auto-clear-sm .col-xs-3:nth-child(4n+1), 
  .auto-clear-sm .col-xs-4:nth-child(3n+1), 
  .auto-clear-sm .col-xs-6:nth-child(odd) {
    clear: none;
  }
  .auto-clear-sm .col-sm-1:nth-child(12n+1), 
  .auto-clear-sm .col-sm-2:nth-child(6n+1), 
  .auto-clear-sm .col-sm-3:nth-child(4n+1), 
  .auto-clear-sm .col-sm-4:nth-child(3n+1), 
  .auto-clear-sm .col-sm-6:nth-child(odd) {
    clear: both;
  }
}
@media (min-width: 992px) {
  .auto-clear-md .col-xs-1:nth-child(12n+1), 
  .auto-clear-md .col-xs-2:nth-child(6n+1), 
  .auto-clear-md .col-xs-3:nth-child(4n+1), 
  .auto-clear-md .col-xs-4:nth-child(3n+1), 
  .auto-clear-md .col-xs-6:nth-child(odd) {
    clear: none;
  }
  .auto-clear-md .col-sm-1:nth-child(12n+1), 
  .auto-clear-md .col-sm-2:nth-child(6n+1), 
  .auto-clear-md .col-sm-3:nth-child(4n+1), 
  .auto-clear-md .col-sm-4:nth-child(3n+1), 
  .auto-clear-md .col-sm-6:nth-child(odd) {
    clear: none;
  }
  .auto-clear-md .col-md-1:nth-child(12n+1), 
  .auto-clear-md .col-md-2:nth-child(6n+1), 
  .auto-clear-md .col-md-3:nth-child(4n+1), 
  .auto-clear-md .col-md-4:nth-child(3n+1), 
  .auto-clear-md .col-md-6:nth-child(odd) {
    clear: both;
  }
}
@media (min-width: 1200px) {
  .auto-clear-lg .col-xs-1:nth-child(12n+1), 
  .auto-clear-lg .col-xs-2:nth-child(6n+1), 
  .auto-clear-lg .col-xs-3:nth-child(4n+1), 
  .auto-clear-lg .col-xs-4:nth-child(3n+1), 
  .auto-clear-lg .col-xs-6:nth-child(odd) {
    clear: none;
  }
  .auto-clear-lg .col-sm-1:nth-child(12n+1), 
  .auto-clear-lg .col-sm-2:nth-child(6n+1), 
  .auto-clear-lg .col-sm-3:nth-child(4n+1), 
  .auto-clear-lg .col-sm-4:nth-child(3n+1), 
  .auto-clear-lg .col-sm-6:nth-child(odd) {
    clear: none;
  }
  .auto-clear-lg .col-md-1:nth-child(12n+1), 
  .auto-clear-lg .col-md-2:nth-child(6n+1), 
  .auto-clear-lg .col-md-3:nth-child(4n+1), 
  .auto-clear-lg .col-md-4:nth-child(3n+1), 
  .auto-clear-lg .col-md-6:nth-child(odd) {
    clear: none;
  }
  .auto-clear-lg .col-lg-1:nth-child(12n+1), 
  .auto-clear-lg .col-lg-2:nth-child(6n+1), 
  .auto-clear-lg .col-lg-3:nth-child(4n+1), 
  .auto-clear-lg .col-lg-4:nth-child(3n+1), 
  .auto-clear-lg .col-lg-6:nth-child(odd) {
    clear: both;
  }
}