Dynamic batching and object local vertex position

I am writing a shader that would pulsate an object by moving vertex position in local coordinates.
It works fine if applied to one object only. However, if it is applied to multiple objects and if they are batched, then local coordinates get messed up and objects pulsate around a common center. Normals are also messed up as can be seen by color. Instead of having a object relative direction, they are relative to a collective of all the batched objects.

 Shader "Custom/Pulsar" {
     Properties {
         _GrowFactor ("Grow Factor", Float) = 0.4
     }
     SubShader {
         Pass {
             Tags { "Queue"="Geometry" }
             Cull Off
             
             CGPROGRAM
             #pragma vertex vert
             #pragma fragment frag
             #include <UnityCG.cginc>
     
             float _GrowFactor;
             
             struct VertexOutput{
                 float4 pos : SV_POSITION;
                 float4 col : COLOR;
             };
             
             VertexOutput vert(appdata_full v){
                 VertexOutput o;
                 float4 vPos = v.vertex + float4(v.normal.xyz, 1.0)*abs(_GrowFactor*_SinTime.w);
                 o.pos = mul(UNITY_MATRIX_MVP, vPos);
                 o.col = float4(v.normal, 1.0);
                 return o;
             }
             
             float4 frag(VertexOutput o): COLOR0 {
                 return o.col;
             }
             
             ENDCG
         }
     } 
     FallBack "Diffuse"
 }

Is there a way that vertices can be processed in their local coordinates even after batching, possibly in a particular transformation space ?