Why mortar the cross webs on either side of a poured cell?

Why do we mortar the cross webs on either side of a poured cell? If the cross webs aren't mortared what can be done after the fact, that is, after the wall is laid up but prior to it being grouted?

The short answer is the obvious answer - the cross webs are mortared to keep the grout from leaking out of the cell being filled. We don't want the grout leaking out for a number of reasons. The important structural reason is that the fluid paste surrounding the pea rock would leak out during the filling and vibrating of the cells leaving the pea rock behind with empty void spaces around the rock at each joint. These void spaces, where the paste is gone and only the rock is left, are commonly know as "honeycombs" in construction. Honeycomb severely weakens hardened grout or concrete where it occurs. This would only apply to course aggregate grout mixes that contain pea rock.

High slump fine aggregate grout mix, that contains only sand and aggregate, is going to simply flow through the joints with little or no segregation between the cement and the sand. This uncontrolled leaking of fine aggregate grout does not present a structural problem as much as a construction problem. The fine aggregate grout will actually fill the adjacent cells, costing the contractor more money for grout.

The contractor will surely be more reluctant to vibrate the grout after initial water loss as the level of fine aggregate grout would continue to drop as the grout leaks out. Unfortunately, this code required final vibration is essential to make sure that grout bridging in the cell has not formed a hollow areas under the bridged area (see below).

Also, the extra grout will increase the weight of the wall. Extra weight, other then in tall multi story buildings, is usually not a problem but should be verified with the designer.

So, to answer the second part of the question directly, if the cross webs are not mortared the wall should be grouted with a fine aggregate grout with the proper slump (8" to 11"). It is essential that this fine aggregate grout be re-consolidated after the initial water loss to collapse any void areas. This will require additional topping off of the wall.
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Section 3.3 B. 4.b.2 of the TMS 602-16 code clearly calls out for the cross webs to be mortared. Fine aggregate grout should not and cannot become a substitute for proper construction.

Links to additional documentation:
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Grouting Concrete Masonry Walls http://ncma-br.org/pdfs/130/TEK%2003-02A.pdf