Modified from James Bruton's sketch example. This sketch uses the serial connection to listen for servo addresses 100 through 115, then activate, pause, and return each servo called.

The Unreal Engine app only has to send the address followed by a newline.

This first sketch had a couple for loops that were too slow:

#include <Wire.h>
#include <Adafruit_PWMServoDriver.h>
Adafruit_PWMServoDriver pwm = Adafruit_PWMServoDriver();
#define SERVOMIN  150 // this is the 'minimum' pulse length count (out of 4096)
#define SERVOMAX  600 // this is the 'maximum' pulse length count (out of 4096)
int action;
long interval = 500;
unsigned long currentMillis;
long previousMillis;
uint8_t servoAddr[] =       {  0,  1,  2,  3,
                               4,  5,  6,  7,
                               8,  9, 10, 11,
                              12, 13, 14, 15  };
                        
int flagAddr[] =            {  0,  0,  0,  0,
                               0,  0,  0,  0,
                               0,  0,  0,  0,
                               0,  0,  0,  0  };
                        
long previousMillisAddr[] = {  0,  0,  0,  0,
                               0,  0,  0,  0,
                               0,  0,  0,  0,
                               0,  0,  0,  0  };
void setup() {
  Serial.begin(9600);
  pwm.begin();
  pwm.setPWMFreq(60);
}
void run_servo(uint8_t &myServo, int &myFlag, long &myPreviousMillis) {
  if (myFlag == 1) {
    // pwm.setPWM(myServo, SERVOMIN, SERVOMAX);
    for (uint16_t pulselen = SERVOMIN; pulselen < SERVOMAX; pulselen++) {
      pwm.setPWM(myServo, 0, pulselen);
    }
    myFlag = 2;
    myPreviousMillis = currentMillis;
  } else if (myFlag == 2 && currentMillis - myPreviousMillis > interval) {
    // pwm.setPWM(myServo, SERVOMAX, SERVOMIN);
    for (uint16_t pulselen = SERVOMAX; pulselen > SERVOMIN; pulselen--) {
      pwm.setPWM(myServo, 0, pulselen);
    }
    myFlag = 0;
    myPreviousMillis = currentMillis;
  }
}
void loop() {
  currentMillis = millis();
  if(currentMillis - previousMillis > interval) {
    previousMillis = currentMillis;
    if (Serial.available() > 0) {
      action = Serial.parseInt();
      if (Serial.read() == '\n') {
        for(uint8_t i = 0; i < 16; i++) {
          if (action == (100 + i)) {
            flagAddr[i] = 1;
          }
        }
      }
    }
    for(uint8_t i = 0; i < 16; i++) {
        run_servo(servoAddr[i], flagAddr[i], previousMillisAddr[i]);
    }
  }
}

Current sketch with the for loops unwound:

#include <Wire.h>
#include <Adafruit_PWMServoDriver.h>
Adafruit_PWMServoDriver pwm = Adafruit_PWMServoDriver();
#define SERVOMIN  150 // this is the 'minimum' pulse length count (out of 4096)
#define SERVOMAX  600 // this is the 'maximum' pulse length count (out of 4096)
int action;
long interval = 2000;
unsigned long currentMillis;
long previousMillis;
uint8_t servoAddr[] =       {  0,  1,  2,  3,
                               4,  5,  6,  7,
                               8,  9, 10, 11,
                              12, 13, 14, 15  };
                        
int flagAddr[] =            {  0,  0,  0,  0,
                               0,  0,  0,  0,
                               0,  0,  0,  0,
                               0,  0,  0,  0  };
                        
long previousMillisAddr[] = {  0,  0,  0,  0,
                               0,  0,  0,  0,
                               0,  0,  0,  0,
                               0,  0,  0,  0  };
void setup() {
  Serial.begin(9600);
  pwm.begin();
  pwm.setPWMFreq(60);
}
void run_servo(uint8_t &myServo, int &myFlag, long &myPreviousMillis) {
  if (myFlag == 1) {
    for (uint16_t pulselen = SERVOMIN; pulselen < SERVOMAX; pulselen++) {
      pwm.setPWM(myServo, 0, pulselen);
    }
    myFlag = 2;
    myPreviousMillis = currentMillis;
  } else if (myFlag == 2 && currentMillis - myPreviousMillis > interval) {
    for (uint16_t pulselen = SERVOMAX; pulselen > SERVOMIN; pulselen--) {
      pwm.setPWM(myServo, 0, pulselen);
    }
    myFlag = 0;
    myPreviousMillis = currentMillis;
  }
}
void loop() {
  currentMillis = millis();
  if(currentMillis - previousMillis > interval) {
    previousMillis = currentMillis;
    if (Serial.available() > 0) {
      action = Serial.parseInt();
      if (Serial.read() == '\n') {
        if (action == (100)) {
          flagAddr[0] = 1;
        }
        if (action == (101)) {
          flagAddr[1] = 1;
        }
        if (action == (102)) {
          flagAddr[2] = 1;
        }
        if (action == (103)) {
          flagAddr[3] = 1;
        }
        if (action == (104)) {
          flagAddr[4] = 1;
        }
        if (action == (105)) {
          flagAddr[5] = 1;
        }
        if (action == (106)) {
          flagAddr[6] = 1;
        }
        if (action == (107)) {
          flagAddr[7] = 1;
        }
        if (action == (108)) {
          flagAddr[8] = 1;
        }
        if (action == (109)) {
          flagAddr[9] = 1;
        }
        if (action == (110)) {
          flagAddr[10] = 1;
        }
        if (action == (111)) {
          flagAddr[11] = 1;
        }
        if (action == (112)) {
          flagAddr[12] = 1;
        }
        if (action == (113)) {
          flagAddr[13] = 1;
        }
        if (action == (114)) {
          flagAddr[14] = 1;
        }
        if (action == (115)) {
          flagAddr[15] = 1;
        }
      }
    }
    run_servo(servoAddr[0], flagAddr[0], previousMillisAddr[0]);
    run_servo(servoAddr[1], flagAddr[1], previousMillisAddr[1]);
    run_servo(servoAddr[2], flagAddr[2], previousMillisAddr[2]);
    run_servo(servoAddr[3], flagAddr[3], previousMillisAddr[3]);
    run_servo(servoAddr[4], flagAddr[4], previousMillisAddr[4]);
    run_servo(servoAddr[5], flagAddr[5], previousMillisAddr[5]);
    run_servo(servoAddr[6], flagAddr[6], previousMillisAddr[6]);
    run_servo(servoAddr[7], flagAddr[7], previousMillisAddr[7]);
    run_servo(servoAddr[8], flagAddr[8], previousMillisAddr[8]);
    run_servo(servoAddr[9], flagAddr[9], previousMillisAddr[9]);
    run_servo(servoAddr[10], flagAddr[10], previousMillisAddr[10]);
    run_servo(servoAddr[11], flagAddr[11], previousMillisAddr[11]);
    run_servo(servoAddr[12], flagAddr[12], previousMillisAddr[12]);
    run_servo(servoAddr[13], flagAddr[13], previousMillisAddr[13]);
    run_servo(servoAddr[14], flagAddr[14], previousMillisAddr[14]);
    run_servo(servoAddr[15], flagAddr[15], previousMillisAddr[15]);
  }
}