Can Kruskal-Wallis test be used to test significance of multiple groups within multiple factors?

I have tried to read what I can on Kruskal-Wallis and while I have found some useful information, I still seem to not find the answer to my question. I am trying to use the Kruskal-Wallis test to determine the significance of multiple groups, within multiple factors, in predicting a set of dependent variables.

Here is an example of my data:

ID   Date     Point   Season  Grazing   Cattle_Type  AvgVOR  PNatGr  NatGrHt  
181 7/21/2015 B22    late     pre       Large         0.8     2        20
182 7/21/2016 B32    early    post      Small         1.0     4        24    

In this example, my dependent variables are "AcgVor", "PNatGR" and"NatGrHt" while the independent variables (factors) are "Season', 'Grazing", and "Cattle_Type". As you can see, each of my factors has 2 group levels each.

What I am trying to accomplish is to run a non-paramatric test that looks at the separate and combined importance of my factor groups to each of my dependent variables. I chose Krukal-Wallis and it seems to work for testing one of my grouping factors at a time.

Here is the result for AvgVor ~ Grazing

 kruskal.test(AvgVOR ~ Grazing, data = Veg)
 Kruskal-Wallis rank sum test
data:  AvgVOR by Grazing
Kruskal-Wallis chi-squared = 94.078, df = 1, p-value < 2.2e-16  

This tells me that AvGVor is significantly different according to whether they were recorded pre or post grazing.

Is there a way to build a similar model using Kruskal-Wallis that includes all of my grouping factors? Even if I have to run a separate one for each dependent variable.

I attempted the following code, but it is flawed.

lapply(Veg[,c("Grazing", "Cattle_Type", "Season")]),function(AvgVOR) kruskal.test(AvgVOR ~ Veg)