Try to use android service with START_STICKY attribute.
In background thread you can listen for location changes (do not use standard LocationSensor - just implement java interface based solution).
You can also find examples of background operations for iOS.
For android you might be interested in unit Androidapi.JNI.Location
TLocationListener = class(TJavaLocal, JLocationListener)
  public
    procedure onLocationChanged(location: JLocation); cdecl;
    procedure onProviderDisabled(provider: JString); cdecl;
    procedure onProviderEnabled(provider: JString); cdecl;
    procedure onStatusChanged(provider: JString; status: Integer; extras: JBundle); cdecl;
end
And for your service module you have to declare some variables 
TServiceModule = class(TAndroidService)
  function AndroidServiceStartCommand(const Sender: TObject;
    const Intent: JIntent; Flags, StartId: Integer): Integer;
private
  FLocationManager: JLocationManager;
  FLocationManagerService: JObject;
  FLocationListener: JLocationListener;
function TServiceModule.AndroidServiceStartCommand(const Sender: TObject;
  const Intent: JIntent; Flags, StartId: Integer): Integer;
begin
  Result := TJService.JavaClass.START_STICKY;
  FLocationManagerService := TAndroidHelper.Context.getSystemService(
    TJContext.JavaClass.LOCATION_SERVICE);
  FLocationManager := TJLocationManager.Wrap(
    (FLocationManagerService as ILocalObject).GetObjectID);
  if FLocationManager.isProviderEnabled(
    TJLocationManager.JavaClass.GPS_PROVIDER) then
  begin
    FLocationListener := TLocationListener.Create;
      FLocationManager.requestLocationUpdates(TJLocationManager.JavaClass.GPS_PROVIDER,
        0, 0, FLocationListener, TJLooper.JavaClass.getMainLooper);