It didn't happen routinely, but there is a quite famous incident at the battle of Manassas (First Bull Run). Washington, DC was only a carriage ride away from where the battle occurred in Northern Virginia and many members of DC high-society, including member of Congress, followed the Union army as it marched off to what was expected to be a quick victory that would nip the rebellion in the bud.

As turned out, of course, the Union suffered a defeat and many of the 'boys in blue' retreated pell-mell back to the Potomac River, and the picnic goers, running away as well, only helped to further hamper the rout. At least one, Congressman Alfred Ely of New York, had the ignominious distinction of being captured.

it must have happened on a few occasions; and probably more so at the beginning of the war. Combat artist Frank VIzetelly sketched southern girls cheering their men into battle in 1862. I think this was at Corinth MIssippi. YOu can see the sketch in the National Geographic magazine APril 1961, along with many other sketches this artist made.

THere's also sketch made of the shelling of Ft. Sumter at the start of the war, with citizens lined up along the sea wall. I think its by Vizetelly, in any event I think there's more than one sketch of that.

You dont hear much about such incidents later in the war. For one thing there were more surprise attacks so the locations of battle were less certain. Also some areas such as central virginia were nearly deserted at the end of the war. But even so in 1864, president Lincoln came out support the troops and was under fire at the battle of Ft Stevens on the ouskirts of Washington.

So yes, it must have certainly happened there is documentation on that. I think it happened more so at the begining of the war when people didnt realize how deadly and how vast this war would become. .