While we have a number of illustrations for the coat there are few extant coats we can look at and of those I believe a number are mislabelled. In the examples below I believe the coat in Figure 1 is an early Ottoman coat based on the fact that the crossover matches that seen in late Ottoman coats. The coat in Figure 2 maybe a late Seljuk coat where the inner and outer crossovers are the same cut or possibly a transitional form between the Late Seljuk and early Ottoman. Figure 3 appears to be the same styling as that in Figure 2, so I'd place that as late Seljuk or early Ottoman.

The coats in Figures 4 - 6 are listed as being from Central Asia and dated roughly around the same period as the other 3 coats. These ones help show the diversity of coat styles that can fall under the term "crossover coats" even within a fairly restricted time period. So based on existing examples the conclusion to draw is that coats can vary widely in this time period and region and that the best we can do is pick one style that meets our needs and proceed with that.

Seljuk Silk Lampas Robe, Central Asia, 11th/12th century
Figure 1 - Seljuk Silk Lampas Robe, Central Asia, 11th/12th century
Seljuk Silk Samite Robe, Central Asia, 11th Century
Figure 2 - Seljuk Silk Samite Robe, Central Asia, 11th Century
Seljuk Silk Lampas Robe, Iran or Central Asia, 11th/12th century
Figure 3 - Seljuk Silk Lampas Robe, Iran or Central Asia, 11th/12th century
Central Asia, probably Kucha, 12th/ 13th Century
Figure 4 - Central Asia, probably Kucha, 12th/ 13th Century
Silk samite. 8th-11th century - uncertain.
Figure 5 - Silk samite. 8th-11th century - uncertain.
A rare silk robe. Central Asia, 11th-12th century
Figure 6 - A rare silk robe. Central Asia, 11th-12th century