I hope I understood your problem. The way I deal with this kind of issues is as follows:

1. query string in URL changes → perform_search and update liveview

2. phx-change event → issue a push_patch with a query string corresponding to the new search input. The push_patch updates the query string and triggers (1)

This way the state of your view, search input and query string in the URL are always in sync.

In (simplified) code:

# This handles (1)
def handle_params(params, _uri, socket) do
    search_input = params["search_field"] <-- This comes from the URL

    {
      :noreply,
      assign(
        socket,
        seach_results: peform_search(search_field)
       )
    }
  end

# This handles (2)
def handle_event(
        "search",
        %{"search_field" => search_field}, <-- This comes from the form input
        socket
      ) do

    {:noreply,
     push_patch(socket,
       to: Routes.api_path(socket, :your_live_view_page_name, search_field: search)
     )}
end 

# This handles the initial mount
def mount(params, _session, socket) do

    {:noreply, socket} = handle_params(params, "", socket)

    {
      :ok,
      socket
    }
  end 

I hope this is what you were looking for.