Was there a time when bishops were not appointed to other sees and did not generally ever relocate?
The simple answer is yes.
In the Early Church, the bishop of a diocese was chosen by the local clergy and not appointed to the diocese by the pope. Thus transfers to other articles sees was extremely rare, if not non-existent.
It was however not unheard of as we see the Prince of the Apostles, St. Peter becoming the Bishop of Rome. Since the Prince of the Apostles, under Divine inspiration, decided to move his see to Rome, it was necessary for him to appoint St. Evodius as his successor of the Diocese of Antioch. The Apostolic Church was simply growing that fast.
It's unclear when the notion of a monarchial bishop emerged, but it is clear that by 200 AD a single bishop in charge of a metropolitan area became a universal norm without much controversy. Initially, bishops were chosen by the local clergy with approval from nearby bishops. "A newly elected bishop was installed in office and given his authority [...] by the bishops who supervised the election and performed the ordination."
Examples of episcopal election in the early church include such notable figures as Ambrose of Milan. Episcopal election was so taken for granted that by the time of the Council of Nicaea, it is mentioned as the normative method for selecting bishops, with approval of local metropolitans.
The bishops of the most important sees sought acceptance from Rome. Some early church fathers attest to the fact that the Church of Rome - in effect its diocese - was the central point of authority. They attest to the Church's reliance on Rome for advice, for mediation of disputes, and for guidance on doctrinal issues. They note, as Ignatius of Antioch (ca 1st century) does, that Rome "holds the presidency" among the other churches, and that, as Irenaeus (ca 2nd century) explains, "because of its superior origin, all the churches must agree" with Rome. They are also clear on the fact that it is full communion with Rome and the bishop of Rome that causes one to be in communion with the Catholic Church. This displays a recognition that, as Cyprian of Carthage (3rd century) puts it, Rome is "the principal church, in which sacerdotal unity has its source." Most of these references were to the entire Church of Rome as such, not necessarily to the Bishop of Rome in his person, but after the role of the pope emerged, the church and its bishop became interpreted in a synonymous way. - Appointment of Catholic bishops
In 325 AD, the Ecumenical Council of Nicaea forbid bishops from being transferred to another see:
Canon 15 - Neither bishop, nor presbyter, nor deacon shall be transferred from city to city. But they shall be sent back should they attempt to do so, to the Churches in which they were ordained.
Slowly, but surely the Sovereign Pontiff took more control over the appointments of bishops and the establishments of dooceses.
By the time of the First Council of Nicaea in 325, the metropolitan bishops of Alexandria, Antioch, and Rome had a role of the greatest importance in the selection. Canon 6 of the Council acknowledged and codified an ancient custom giving jurisdiction over large regions to the bishops of Alexandria, Rome, and Antioch. Nicaea decreed that the consent of the metropolitan bishop was normally required:
Let the ancient customs in Egypt, Libya and Pentapolis prevail, that the Bishop of Alexandria have jurisdiction in all these, since the like is customary for the Bishop of Rome also. Likewise in Antioch and the other provinces, let the Churches retain their privileges. And this is to be universally understood, that if any one be made bishop without the consent of the Metropolitan, the great Synod has declared that such a man ought not to be a bishop. If, however, two or three bishops shall from natural love of contradiction, oppose the common suffrage of the rest, it being reasonable and in accordance with the ecclesiastical law, then let the choice of the majority prevail. - Appointment of Catholic bishops
In the 11th century the pope reserved to himself the creation of new dioceses.
We have noticed above that after the eleventh century the sovereign pontiff reserved to himself the creation of dioceses. In the actual discipline, as already stated, all that touches the diocese is a causa major, i.e. one of those important matters in which the bishop possesses no authority whatever and which the pope reserves exclusively to himself. Since the episcopate is of Divine institution, the pope is obliged to establish dioceses in the Catholic Church, but he remains sole judge of the time and manner, and alone determines what flock shall be entrusted to each bishop. - Diocese (Catholic Encyclopedia)
However it was not until the the 19th century that the Holy See took control over the nominations of bishops and the possibility of episcopal jurisdiction transfer of a bishop from one diocese to another.
Centralization of papal power
In the early 19th century, state involvement in episcopal appointment was still so normal that, in spite of the opposition of the Church in Ireland to the proposed royal veto of the appointment of bishops, the Holy See was prepared to grant it to the British king. As late as the 20th century, Franz Joseph I of Austria-Hungary exercised a power of Jus exclusivae to veto the election of Mariano Rampolla as pope during the 1903 papal conclave. After Rampolla was vetoed, the conclave elected Pope Pius X.
It was in 1871 that a radical shift in law and practice began to take place. In that year the Law of guarantees gave the pope the right to choose the bishops of the Kingdom of Italy, all 237 of them, appointments that through the unification of Italy had fallen into the hands of King Victor Emmanuel II of Italy. Although the pope denounced the law, he nevertheless profited by it to appoint, within the first seven months that followed, 102 new Italian bishops. Before the unification of Italy the various rulers made the appointments, with the pope doing so only for the Papal States. The 1905 French law on the Separation of the Churches and the State had a similar effect for appointing bishops in the territories ruled by France. In the early 20th century papal appointment of Catholic bishops was an almost universal practice except where, in virtue of the Spanish patronato real and the Portuguese padroado, the appointment of Catholic bishops remained in the hands of the civil authorities.
Thus the 1917 Code of Canon Law was able to finally affirm that, in the Latin Church, the decision rests with the pope. In the course of the 20th century, remaining privileges enjoyed by secular authorities gradually diminished, especially since the Second Vatican Council (1962–1965), which declared that the right of nominating and appointing bishops belongs per se exclusively to the competent ecclesiastical authority and requested the civil authorities who still had rights and privileges in this field to renounce them voluntarily. - Appointment of Catholic bishops
As a general rule, since Vatican II, archbishops are always chosen from diocesan bishops of the country in question or ecclesiastical region. I can recall, when Bishop Exner was appointed Archbishop of Vancouver and everyone thought it amazing that I predicted his appoint to this see. The fact was much more simple. I followed the logical conclusion that Rome might make. It is not magic, but simply logic that made the intuition possible.
Bishops are still considered to be married to their diocese that is the Church which is the Bride of Christ mystically speaking. Some rules for bishops point this out:
Must live in the diocese for most of the year, unless on official business in Rome or when attending a Church Council.
Has a serious duty to be present at his cathedral during the seasons of Lent and Advent, and on the feasts of Easter, Pentecost, Christmas and Corpus Christi.
Every five years a bishop must send a report to Rome.
Bishops must visit Rome to meet with the Pope at least once every five years and visit the tombs of the Apostles Peter and Paul.
Bishops must regularly visit each part of their diocese.
Bishops must attend Synods (meetings of Bishops) when they are called. - The responsibilities and role of a bishop