Did God promise to appoint prophets in each generation? I remember a TV show where one of the characters tells another person that God promised that there would be prophets in each generation. Is there Biblical support for such a view? If this view does exist what are some of the Bible verses that support? I'm interested in the main Catholic and Protestant view on this issue.
Introduction

A prophet is?


  A delegate or mouthpiece of God - cf. PROPHETS AND PROPHECY |  Jewish
  Encyclopedia.


From the said Jewish Encyclopedia above, 'prophecy was not regarded as confined to Israel.'

And from this selected answer | Mi Yodeya


  Once the last prophets -- Haggai, Zechariah and Malachi -- died, the
  prophetic spirit ceased in Israel.


Therefore, at least from this answer, Israel who had Moses, have not had the prophetic spirit in the generations after the last three prophets mentioned above.

In the New Testament St. Paul makes it clear that Jesus gifts his Church with prophets for building up the body of Christ which is the Church


  Eph 4:11-14 (RSVCE) 11 And his gifts were that some should be
  apostles, some prophets, some evangelists, some pastors and teachers,
  12 for the equipment of the saints, for the work of ministry, for building up the body of Christ, 13 until we all attain to the
  unity of the faith and of the knowledge of the Son of God, to mature
  manhood, to the measure of the stature of the fulness of Christ;
  14 so that we may no longer be children, tossed to and fro and carried about with every wind of doctrine, by the cunning of men, by
  their craftiness in deceitful wiles.


And the Catholic Church teaches that all the baptized share in the priesthood of Christ, in his prophetic and royal mission


  CCC 1268 The baptized have become "living stones" to be
  "built into a spiritual house, to be a holy priesthood."1
  By Baptism they share in the priesthood of Christ, in his prophetic
  and royal mission. They are "a chosen race, a royal priesthood, a holy
  nation, God's own people, that [they] may declare the wonderful deeds
  of him who called [them] out of darkness into his marvelous
  light."2 Baptism gives a share in the common priesthood of
  all believers.
  
  1. 1 Pet 2:5.
  2. 1 Pet 2:9.


One can safely conclude that the baptized here on C.SE are prophets (mouthpieces of God) provided their contributions are faithful to the Church's Teaching.