In Anglicanism, processions are probably most popular among Anglo-Catholics, and indeed historically the Church of England went without processions within church buildings (except entrance processions) for quite some time.
But this relative lack of processions is due more to accident of history than to a reforming plan, and certainly there is no Reformed principle against walking to pray or praying while walking. If we want to restore vigil worship and instruction on the nights before feast days, processions may well form a part of that plan. The point is not to instruct the people by reason alone, but also to bring along their bodies into remembrance of what God has done and will do, and thus to build up good Christian cultural habits to leaven a whole people.
According to Sally Elizabeth (Roper) Harper, writing in Medieval English Benedictine Liturgy: Studies in the Formation, Structure, and Content of the Monastic Votive Office, c. 950–1540,
Most processions consisted of three stages: in eundo, where the procession left the choir, a focal statio, or station, and finally in redeundo, the return to choir. The outward phase was normally accompanied by a borrowed respond, or sometimes a proper processional antiphon or hymn. At its destination (an altar or chapel, the rood or some other focal point), the procession stopped for a versicle and collect, sometimes said with an antiphon. The third stage of the procession, in redeundo, was effectively a self-contained memorial. An antiphon (or less often a respond) was sung as the procession moved off from the station, usually halting for the versicle and collect at the choir-step. Often the independence of this final stage was emphasized in that it honoured a quite different saint or intention from the preceding parts of the procession (quite commonly the patron).
The first two of these stages are fairly straightforward. In eundo, the procession would leave the church chancel and sing a hymn or anthem as it went. Arriving at its destination, in statione, the procession would stop for a versicle and collect. For example, at the end of Evensong on Christmas Day, the procession would remember St Stephen for the next day. Leaving the chancel, the procession might be accompanied by the hymn ‘Saint of God, Elect and Precious’.
The procession would stop at its station, some place where St Stephen would most readily come to mind:
V. Behold, I see the heavens opened.
R. And the Son of man standing on the right hand of God.
Grant, O Lord, that in all our sufferings here upon earth, for the testimony of thy truth, we may stedfastly look up to heaven, and by faith behold the glory that shall be revealed; and, being filled with the Holy Ghost, may learn to love and bless our persecutors by the example of thy first Martyr Saint Stephen, who prayed for his murderers to thee, O blessed Jesus, who standest at the right hand of God to succour all those that suffer for thee, our only Mediator and Advocate. Amen.
Upon the return, if the procession was not already in commemoration of the saint after whom the church was named, then a fitting anthem commemorating that titular saint would be sung. For example, in a Reformed cathedral named after the Blessed Virgin Mary, an acceptable Marian anthem would be sung, perhaps one from the Song of Songs, directing prayer toward no one but God. These words by Thomas Ken could also fit quite well:
1. Her Virgin eyes saw God incarnate born,
When she to Bethl’em came that happy morn;
How high her raptures then began to swell,
None but her own omniscient Son can tell.
2. As Eve when she her fontal sin reviewed,
Wept for herself and all she should include,
Blest Mary with man’s Saviour in embrace
Joyed for herself and for all human race.
3. All saints are by her Son’s dear influence blest,
She kept the very Fountain at her breast;
The Son adored and nursed by the sweet Maid
A thousandfold of love for love repaid.
4. Heaven with transcendent joys her entrance graced,
Next to his throne her Son his Mother placed;
And here below, now she’s of heaven possest,
All generations are to call her blest.
Then would follow the Collect for the Annunciation, with a versicle and response before it.
V. Hail Mary, full of grace, the Lord is with thee.
R. Blessed art thou amongst women, and blessed is the fruit of thy womb, Jesus.
Let us pray.
We beseech thee, O Lord, pour thy grace into our hearts; that, as we have known the incarnation of thy Son Jesus Christ by the message of an angel, so by his cross and passion we may be brought unto the glory of his resurrection; through the same Jesus Christ our Lord. Amen.
But if the procession was already in commemoration of the Holy Virgin rather than St Stephen, then instead in redeundo an anthem should be sung for the Feast of All Saints, and then the collect following with its versicle and response before it.
V. Be glad, O ye righteous, and rejoice in the Lord.
R. And be joyful, all ye that are true of heart.
Let us pray.
O almighty God, who hast knit together thine elect in one communion and fellowship, in the mystical body of thy Son Christ our Lord: Grant us grace so to follow thy blessed Saints in all virtuous and godly living, that we may come to those unspeakable joys, which thou hast prepared for them that unfeignedly love thee; through Jesus Christ our Lord. Amen.
From Easter Day to Ascension, the anthem in redeundo would be better replaced with one about the Resurrection, perhaps to one of those which the Prayer Book orders for Morning Prayer on Easter Day.
V. The Lord is risen from the grave.
R. Who hung for us upon the tree.
Let us pray.
Almighty God, whose beloved Son for our sake willingly offered himself to endure the agony and shame of the Cross: Remove from us all cowardice of heart, and give us courage to take up our cross and bear it patiently in his service; through the same thy Son Jesus Christ our Lord, who liveth and reigneth with thee and the Holy Ghost ever, one God, world without end. Amen.