The question as it is written is a bit unclear, however, I'll attempt to demonstrate a solution based on my understanding.
First off, I'm unsure what you are trying to accomplish with the function multirow_spec as the package function row_spec is already vectorised for the option row... though not for background.
Second, I'm assuming that you want the colours to vary based on the previous value in the same column (gear) as that is the only one mentioned in the question. We will modify the column using mutate, case_when, and lag all from dplyr. We'll use cell_spec to target the individual cell values.
library(dplyr)
library(knitr)
library(kableExtra)
mtcars %>%
slice(1:10) %>%
mutate(
gear = case_when(
gear > lag(gear) ~ cell_spec(gear, background = "green"),
gear == lag(gear) ~ cell_spec(gear, background = "yellow"),
gear < lag(gear) ~ cell_spec(gear, background = "red")
)) %>%
kable("html", escape = FALSE)
N.B. The option escape = FALSE is required so that the html content generated with cell_spec is not escaped.
<table>
<thead>
<tr>
<th style="text-align:right;"> mpg </th>
<th style="text-align:right;"> cyl </th>
<th style="text-align:right;"> disp </th>
<th style="text-align:right;"> hp </th>
<th style="text-align:right;"> drat </th>
<th style="text-align:right;"> wt </th>
<th style="text-align:right;"> qsec </th>
<th style="text-align:right;"> vs </th>
<th style="text-align:right;"> am </th>
<th style="text-align:left;"> gear </th>
<th style="text-align:right;"> carb </th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:right;"> 21.0 </td>
<td style="text-align:right;"> 6 </td>
<td style="text-align:right;"> 160.0 </td>
<td style="text-align:right;"> 110 </td>
<td style="text-align:right;"> 3.90 </td>
<td style="text-align:right;"> 2.620 </td>
<td style="text-align:right;"> 16.46 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:left;"> NA </td>
<td style="text-align:right;"> 4 </td>
</tr>
<tr>
<td style="text-align:right;"> 21.0 </td>
<td style="text-align:right;"> 6 </td>
<td style="text-align:right;"> 160.0 </td>
<td style="text-align:right;"> 110 </td>
<td style="text-align:right;"> 3.90 </td>
<td style="text-align:right;"> 2.875 </td>
<td style="text-align:right;"> 17.02 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:left;"> <span style=" border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: yellow;">4</span> </td>
<td style="text-align:right;"> 4 </td>
</tr>
<tr>
<td style="text-align:right;"> 22.8 </td>
<td style="text-align:right;"> 4 </td>
<td style="text-align:right;"> 108.0 </td>
<td style="text-align:right;"> 93 </td>
<td style="text-align:right;"> 3.85 </td>
<td style="text-align:right;"> 2.320 </td>
<td style="text-align:right;"> 18.61 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:left;"> <span style=" border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: yellow;">4</span> </td>
<td style="text-align:right;"> 1 </td>
</tr>
<tr>
<td style="text-align:right;"> 21.4 </td>
<td style="text-align:right;"> 6 </td>
<td style="text-align:right;"> 258.0 </td>
<td style="text-align:right;"> 110 </td>
<td style="text-align:right;"> 3.08 </td>
<td style="text-align:right;"> 3.215 </td>
<td style="text-align:right;"> 19.44 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:left;"> <span style=" border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: red;">3</span> </td>
<td style="text-align:right;"> 1 </td>
</tr>
<tr>
<td style="text-align:right;"> 18.7 </td>
<td style="text-align:right;"> 8 </td>
<td style="text-align:right;"> 360.0 </td>
<td style="text-align:right;"> 175 </td>
<td style="text-align:right;"> 3.15 </td>
<td style="text-align:right;"> 3.440 </td>
<td style="text-align:right;"> 17.02 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:left;"> <span style=" border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: yellow;">3</span> </td>
<td style="text-align:right;"> 2 </td>
</tr>
<tr>
<td style="text-align:right;"> 18.1 </td>
<td style="text-align:right;"> 6 </td>
<td style="text-align:right;"> 225.0 </td>
<td style="text-align:right;"> 105 </td>
<td style="text-align:right;"> 2.76 </td>
<td style="text-align:right;"> 3.460 </td>
<td style="text-align:right;"> 20.22 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:left;"> <span style=" border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: yellow;">3</span> </td>
<td style="text-align:right;"> 1 </td>
</tr>
<tr>
<td style="text-align:right;"> 14.3 </td>
<td style="text-align:right;"> 8 </td>
<td style="text-align:right;"> 360.0 </td>
<td style="text-align:right;"> 245 </td>
<td style="text-align:right;"> 3.21 </td>
<td style="text-align:right;"> 3.570 </td>
<td style="text-align:right;"> 15.84 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:left;"> <span style=" border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: yellow;">3</span> </td>
<td style="text-align:right;"> 4 </td>
</tr>
<tr>
<td style="text-align:right;"> 24.4 </td>
<td style="text-align:right;"> 4 </td>
<td style="text-align:right;"> 146.7 </td>
<td style="text-align:right;"> 62 </td>
<td style="text-align:right;"> 3.69 </td>
<td style="text-align:right;"> 3.190 </td>
<td style="text-align:right;"> 20.00 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:left;"> <span style=" border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: green;">4</span> </td>
<td style="text-align:right;"> 2 </td>
</tr>
<tr>
<td style="text-align:right;"> 22.8 </td>
<td style="text-align:right;"> 4 </td>
<td style="text-align:right;"> 140.8 </td>
<td style="text-align:right;"> 95 </td>
<td style="text-align:right;"> 3.92 </td>
<td style="text-align:right;"> 3.150 </td>
<td style="text-align:right;"> 22.90 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:left;"> <span style=" border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: yellow;">4</span> </td>
<td style="text-align:right;"> 2 </td>
</tr>
<tr>
<td style="text-align:right;"> 19.2 </td>
<td style="text-align:right;"> 6 </td>
<td style="text-align:right;"> 167.6 </td>
<td style="text-align:right;"> 123 </td>
<td style="text-align:right;"> 3.92 </td>
<td style="text-align:right;"> 3.440 </td>
<td style="text-align:right;"> 18.30 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:left;"> <span style=" border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: yellow;">4</span> </td>
<td style="text-align:right;"> 4 </td>
</tr>
</tbody>
</table>