The rite in use among the Carmelites since about the middle of the twelfth century is known by the name of the Rite of the Holy Sepulchre.
This Rite of the Holy Sepulchre belonged to the Gallican family of the Roman Rite; it appears to have descended directly from the Parisian Rite, but to have undergone some modifications pointing to other sources.
The reform of the Roman liturgical books under St. Pius V called for a corresponding reform of the Carmelite Rite, which was taken in hand in 1580, the Breviary appearing in 1584 and the Missal in 1587. At the same time the Holy See withdrew the right hitherto exercised by the chapters and the generals of altering the liturgy of the order, and placed all such matters in the hands of the Sacred Congregation of Rites.
A note on the Discalced Carmelites
The publication of the Reformed Breviary of 1584 caused the newly established Discalced Carmelites to abandon the ancient rite once for all and to adopt the Roman Rite instead [NLM note: which breviary I have seen titled the Breviarium Romano-Carmelitanum, which is not to be confused with the Carmelite rite. I believe they also opted to use the Missale Romanum rather than the ancient Carmelite rite.]
The Carmelite Missal
The ancient Carmelite Rite stands about half way between the Carthusian and the Dominican rites. It shows signs of great antiquity -- e.g. in the absence of liturgical colours, in the sparing use of altar candles (one at low Mass, none on the altar itself at high Mass but only acolytes' torches, even these being extinguished during part of the Mass, four torches and one candle in choir for Tenebræ); incense is also used rarely and with noteworthy restrictions; the Blessing at the end of the Mass is only permitted where the custom of the country requires it; passing before the tabernacle, the brethren must make a profound inclination, not a genuflexion.
In the Mass there are some peculiarities. the altar remains covered until the priest and ministers are ready to begin, when the acolytes then roll back the cover; before the end of the Mass they cover the altar again. On great feasts the Introit is said three times, i.e. it is repeated both before and after the Gloria Patri; besides the Epistle and Gospel there is a lesson or prophecy to be recited by an acolyte. At the Lavabo the priest leaves the altar for the piscina where he says that psalm, or else Veni Creator Spiritus or Deus misereatur. Likewise after the first ablution he goes to the piscina to wash his fingers. During the Canon of the Mass the deacon moves a fan to keep the flies away, a custom still in use in Sicily and elsewhere. At the word fregit in the form of consecration, according to the Ordinal of 1312 and later rubrics, the priest makes a movement as if breaking the host. Great care is taken that the smoke of the thurible and of the torches do not interfere with the clear vision of the host when lifted up for the adoration of the faithful but the chalice is only slightly elevated. The celebrating priest does not genuflect but bows reverently. After the Pater Noster the choir sings the psalm Deus venerunt genies for the restoration of the Holy Land. The prayers for communion are identical with those of the Sarum Rite and other similar uses, viz. domine sancte pater, Domine Jesu Christe (as in the Roman Rite), and Salve salus mundi. The Domine non sum dignus was introduced only in 1568. The Mass ended with Dominus vobiscum, Ite missa est (or its equivalent) and Placeat.
The Last Gospel, which in both ordinals serves for the priest's thanksgiving, appears in the Missal of 1490 as an integral part of the Mass.
The Carmelite Breviary
The Divine Office also presents some noteworthy features. The first Vespers of certain feasts and the Vespers during Lent have a responsory usually taken from Matins. Compline has various hymns according to the season, and also special antiphons for the Canticle. The lessons at Matins follow a somewhat different plan from those of the Roman Office. The singing of the genealogies of Christ after Matins on Christmas and the Epiphany gave rise to beautiful ceremonies. After Tenebræ in Holy Week (sung at midnight) comes the chant of the Tropi; all the Holy Week services present interesting archaic features. Other particularities are the antiphons Pro fidei meritis etc. on the Sundays from Trinity Sunday to Advent and the verses after the psalms on Trinity, the feasts of St. Paul and St. Laurence. The hymns are those of the Roman Office; the proses appear to be a uniform collection which remained practically unchanged from the thirteenth century to 1544, when all but four or five were abolished. The Ordinal prescribes only four processions in the course of the year: on Candlemas, Palm Sunday, the Ascension and the Assumption.
The calendar of saints, in the two oldest recensions of the Ordinal, exhibits some feasts proper to the Holy Land, namely some of the early bishops of Jerusalem, the Biblical Patriarchs Abraham, Isaac and Jacob, and Lazarus. The only special features were the feast of St. Anne, probably due to the fact that the Carmelites occupied for a short time a convent dedicated to her in Jerusalem (vacated by Benedictine nuns at the capture of that city in 1187), and the octave of the Nativity of Our Lady, which also was proper to the order
DISCALCED CARMELITE PROPER OFFICES OF CARMELITE SAINTS.
https://ocdssacramento.org/wp-content/uploads/2019/07/office2007.pdf
Original Carmelite Breviary 1539
https://drive.google.com/file/d/1I5QjbzSNaGgtkRxaSN338w5dgSUx9uLn/view?usp=drivesdk