I havent used "basic alumina" columns but silica gel is acidic (OH on the surface) so just based on that I would guess not. Plus anytime you change the stationary phase you are going to get different Rf.
Basically elution pattern of compounds from the mixture depends on nature of the mobile phase (polar or nonpolar) and column packing material. In case of alumina i.e. basic in nature, the compound having acidic character eluted first then so on. But silica is acidic in character. By changing the ratio of mobile phase you also get different Rf value in same St. phase column.
The silicagel is similar to Al2O3 and its absorbic abiliies are closelly, but there are "some little differences", so the picture in SiO2 TLC would differ from the "picture" on Al2O3 column. But in the first approximation the pictures are similar and if you separate two products, one runs fast and the second as a tortoise, then the control is quite suitable
No- alumina has different "selectivity" compared to silica. The interactions between the compound and alumina is usually a different mechanism compared to that of silica.
In the poster reference below, alumina and silica columns were run with the same solvents on the same sample, but the elution is very different for the two columns: